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Properties

Label 256.1
Order \( 2^{8} \)
Exponent \( 2^{8} \)
Abelian yes
$\card{\Aut(G)}$ \( 2^{7} \)
Perm deg. $256$
Trans deg. $256$
Rank $1$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := CyclicGroup(256);
 
Copy content gap:G := CyclicGroup(256);
 
Copy content sage:G = CyclicPermutationGroup(256)
 
Copy content sage_gap:G = libgap.SmallGroup(256, 1)
 
Copy content comment:Define the group as a permutation group
 

Group information

Description:$C_{256}$
Order: \(256\)\(\medspace = 2^{8} \)
Copy content comment:Order of the group
 
Copy content magma:Order(G);
 
Copy content gap:Order(G);
 
Copy content sage:G.order()
 
Copy content sage_gap:G.Order()
 
Exponent: \(256\)\(\medspace = 2^{8} \)
Copy content comment:Exponent of the group
 
Copy content magma:Exponent(G);
 
Copy content gap:Exponent(G);
 
Copy content sage:G.exponent()
 
Copy content sage_gap:G.Exponent()
 
Automorphism group:$C_2\times C_{64}$, of order \(128\)\(\medspace = 2^{7} \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_2$ x 8
Copy content comment:Composition factors of the group
 
Copy content magma:CompositionFactors(G);
 
Copy content gap:CompositionSeries(G);
 
Copy content sage:G.composition_series()
 
Copy content sage_gap:G.CompositionSeries()
 
Nilpotency class:$1$
Copy content comment:Nilpotency class of the group
 
Copy content magma:NilpotencyClass(G);
 
Copy content gap:NilpotencyClassOfGroup(G);
 
Copy content sage_gap:G.NilpotencyClassOfGroup()
 
Derived length:$1$
Copy content comment:Derived length of the group
 
Copy content magma:DerivedLength(G);
 
Copy content gap:DerivedLength(G);
 
Copy content sage_gap:G.DerivedLength()
 

This group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.

Copy content comment:Determine if the group G is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content gap:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content sage_gap:G.IsAbelian()
 
Copy content comment:Determine if the group G is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content gap:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content sage_gap:G.IsCyclic()
 
Copy content comment:Determine if the group G is nilpotent
 
Copy content magma:IsNilpotent(G);
 
Copy content gap:IsNilpotentGroup(G);
 
Copy content sage:G.is_nilpotent()
 
Copy content sage_gap:G.IsNilpotentGroup()
 
Copy content comment:Determine if the group G is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content gap:IsSolvableGroup(G);
 
Copy content sage:G.is_solvable()
 
Copy content sage_gap:G.IsSolvableGroup()
 
Copy content comment:Determine if the group G is supersolvable
 
Copy content gap:IsSupersolvableGroup(G);
 
Copy content sage:G.is_supersolvable()
 
Copy content sage_gap:G.IsSupersolvableGroup()
 
Copy content comment:Determine if the group G is simple
 
Copy content magma:IsSimple(G);
 
Copy content gap:IsSimpleGroup(G);
 
Copy content sage_gap:G.IsSimpleGroup()
 

Group statistics

Copy content comment:Compute statistics for the group G
 
Copy content magma:// Magma code to output the first two rows of the group statistics table element_orders := [Order(g) : g in G]; orders := Set(element_orders); printf "Orders: %o\n", orders; printf "Elements: %o %o\n", [#[x : x in element_orders | x eq n] : n in orders], Order(G); cc_orders := [cc[1] : cc in ConjugacyClasses(G)]; printf "Conjugacy classes: %o %o\n", [#[x : x in cc_orders | x eq n] : n in orders], #cc_orders;
 
Copy content gap:# Gap code to output the first two rows of the group statistics table element_orders := List(Elements(G), g -> Order(g)); orders := Set(element_orders); Print("Orders: ", orders, "\n"); element_counts := List(orders, n -> Length(Filtered(element_orders, x -> x = n))); Print("Elements: ", element_counts, " ", Size(G), "\n"); cc_orders := List(ConjugacyClasses(G), cc -> Order(Representative(cc))); cc_counts := List(orders, n -> Length(Filtered(cc_orders, x -> x = n))); Print("Conjugacy classes: ", cc_counts, " ", Length(ConjugacyClasses(G)), "\n");
 
Copy content sage:# Sage code to output the first two rows of the group statistics table element_orders = [g.order() for g in G] orders = sorted(list(set(element_orders))) print("Orders:", orders) print("Elements:", [element_orders.count(n) for n in orders], G.order()) cc_orders = [cc[0].order() for cc in G.conjugacy_classes()] print("Conjugacy classes:", [cc_orders.count(n) for n in orders], len(cc_orders))
 

Order 1 2 4 8 16 32 64 128 256
Elements 1 1 2 4 8 16 32 64 128 256
Conjugacy classes   1 1 2 4 8 16 32 64 128 256
Divisions 1 1 1 1 1 1 1 1 1 9
Autjugacy classes 1 1 1 1 1 1 1 1 1 9

Copy content comment:Compute statistics about the characters of G
 
Copy content magma:// Outputs [<d_1,c_1>, <d_2,c_2>, ...] where c_i is the number of irr. complex chars. of G with degree d_i CharacterDegrees(G);
 
Copy content gap:# Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i CharacterDegrees(G);
 
Copy content sage:# Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i character_degrees = [c[0] for c in G.character_table()] [[n, character_degrees.count(n)] for n in set(character_degrees)]
 
Copy content sage_gap:G.CharacterDegrees()
 

Dimension 1 2 4 8 16 32 64 128
Irr. complex chars.   256 0 0 0 0 0 0 0 256
Irr. rational chars. 2 1 1 1 1 1 1 1 9

Minimal presentations

Permutation degree:$256$
Transitive degree:$256$
Rank: $1$
Inequivalent generators: $1$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 1 2 128
Arbitrary 1 2 128

Constructions

Show commands: Gap / Magma / SageMath


Presentation: $\langle a \mid a^{256}=1 \rangle$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([8, -2, -2, -2, -2, -2, -2, -2, -2, 16, 41, 66, 91, 116, 141, 166]); a := Explode([G.1]); AssignNames(~G, ["a", "a2", "a4", "a8", "a16", "a32", "a64", "a128"]);
 
Copy content gap:G := PcGroupCode(9747559364451795873038463,256); a := G.1;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(9747559364451795873038463,256)'); a = G.1;
 
Copy content sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(9747559364451795873038463,256)'); a = G.1;
 
Permutation group:Degree $256$ $\langle(1,256,128,192,64,224,96,160,32,240,112,176,48,208,80,144,16,248,120,184,56,216,88,152,24,232,104,168,40,200,72,136,8,252,124,188,60,220,92,156,28,236,108,172,44,204,76,140,12,244,116,180,52,212,84,148,20,228,100,164,36,196,68,132,4,254,126,190,62,222,94,158,30,238,110,174,46,206,78,142,14,246,118,182,54,214,86,150,22,230,102,166,38,198,70,134,6,250,122,186,58,218,90,154,26,234,106,170,42,202,74,138,10,242,114,178,50,210,82,146,18,226,98,162,34,194,66,130,2,255,127,191,63,223,95,159,31,239,111,175,47,207,79,143,15,247,119,183,55,215,87,151,23,231,103,167,39,199,71,135,7,251,123,187,59,219,91,155,27,235,107,171,43,203,75,139,11,243,115,179,51,211,83,147,19,227,99,163,35,195,67,131,3,253,125,189,61,221,93,157,29,237,109,173,45,205,77,141,13,245,117,181,53,213,85,149,21,229,101,165,37,197,69,133,5,249,121,185,57,217,89,153,25,233,105,169,41,201,73,137,9,241,113,177,49,209,81,145,17,225,97,161,33,193,65,129) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 256 | (1,256,128,192,64,224,96,160,32,240,112,176,48,208,80,144,16,248,120,184,56,216,88,152,24,232,104,168,40,200,72,136,8,252,124,188,60,220,92,156,28,236,108,172,44,204,76,140,12,244,116,180,52,212,84,148,20,228,100,164,36,196,68,132,4,254,126,190,62,222,94,158,30,238,110,174,46,206,78,142,14,246,118,182,54,214,86,150,22,230,102,166,38,198,70,134,6,250,122,186,58,218,90,154,26,234,106,170,42,202,74,138,10,242,114,178,50,210,82,146,18,226,98,162,34,194,66,130,2,255,127,191,63,223,95,159,31,239,111,175,47,207,79,143,15,247,119,183,55,215,87,151,23,231,103,167,39,199,71,135,7,251,123,187,59,219,91,155,27,235,107,171,43,203,75,139,11,243,115,179,51,211,83,147,19,227,99,163,35,195,67,131,3,253,125,189,61,221,93,157,29,237,109,173,45,205,77,141,13,245,117,181,53,213,85,149,21,229,101,165,37,197,69,133,5,249,121,185,57,217,89,153,25,233,105,169,41,201,73,137,9,241,113,177,49,209,81,145,17,225,97,161,33,193,65,129), (1,128,64,96,32,112,48,80,16,120,56,88,24,104,40,72,8,124,60,92,28,108,44,76,12,116,52,84,20,100,36,68,4,126,62,94,30,110,46,78,14,118,54,86,22,102,38,70,6,122,58,90,26,106,42,74,10,114,50,82,18,98,34,66,2,127,63,95,31,111,47,79,15,119,55,87,23,103,39,71,7,123,59,91,27,107,43,75,11,115,51,83,19,99,35,67,3,125,61,93,29,109,45,77,13,117,53,85,21,101,37,69,5,121,57,89,25,105,41,73,9,113,49,81,17,97,33,65)(129,256,192,224,160,240,176,208,144,248,184,216,152,232,168,200,136,252,188,220,156,236,172,204,140,244,180,212,148,228,164,196,132,254,190,222,158,238,174,206,142,246,182,214,150,230,166,198,134,250,186,218,154,234,170,202,138,242,178,210,146,226,162,194,130,255,191,223,159,239,175,207,143,247,183,215,151,231,167,199,135,251,187,219,155,235,171,203,139,243,179,211,147,227,163,195,131,253,189,221,157,237,173,205,141,245,181,213,149,229,165,197,133,249,185,217,153,233,169,201,137,241,177,209,145,225,161,193), (1,64,32,48,16,56,24,40,8,60,28,44,12,52,20,36,4,62,30,46,14,54,22,38,6,58,26,42,10,50,18,34,2,63,31,47,15,55,23,39,7,59,27,43,11,51,19,35,3,61,29,45,13,53,21,37,5,57,25,41,9,49,17,33)(65,128,96,112,80,120,88,104,72,124,92,108,76,116,84,100,68,126,94,110,78,118,86,102,70,122,90,106,74,114,82,98,66,127,95,111,79,119,87,103,71,123,91,107,75,115,83,99,67,125,93,109,77,117,85,101,69,121,89,105,73,113,81,97)(129,192,160,176,144,184,152,168,136,188,156,172,140,180,148,164,132,190,158,174,142,182,150,166,134,186,154,170,138,178,146,162,130,191,159,175,143,183,151,167,135,187,155,171,139,179,147,163,131,189,157,173,141,181,149,165,133,185,153,169,137,177,145,161)(193,256,224,240,208,248,216,232,200,252,220,236,204,244,212,228,196,254,222,238,206,246,214,230,198,250,218,234,202,242,210,226,194,255,223,239,207,247,215,231,199,251,219,235,203,243,211,227,195,253,221,237,205,245,213,229,197,249,217,233,201,241,209,225), (1,32,16,24,8,28,12,20,4,30,14,22,6,26,10,18,2,31,15,23,7,27,11,19,3,29,13,21,5,25,9,17)(33,64,48,56,40,60,44,52,36,62,46,54,38,58,42,50,34,63,47,55,39,59,43,51,35,61,45,53,37,57,41,49)(65,96,80,88,72,92,76,84,68,94,78,86,70,90,74,82,66,95,79,87,71,91,75,83,67,93,77,85,69,89,73,81)(97,128,112,120,104,124,108,116,100,126,110,118,102,122,106,114,98,127,111,119,103,123,107,115,99,125,109,117,101,121,105,113)(129,160,144,152,136,156,140,148,132,158,142,150,134,154,138,146,130,159,143,151,135,155,139,147,131,157,141,149,133,153,137,145)(161,192,176,184,168,188,172,180,164,190,174,182,166,186,170,178,162,191,175,183,167,187,171,179,163,189,173,181,165,185,169,177)(193,224,208,216,200,220,204,212,196,222,206,214,198,218,202,210,194,223,207,215,199,219,203,211,195,221,205,213,197,217,201,209)(225,256,240,248,232,252,236,244,228,254,238,246,230,250,234,242,226,255,239,247,231,251,235,243,227,253,237,245,229,249,233,241), (1,16,8,12,4,14,6,10,2,15,7,11,3,13,5,9)(17,32,24,28,20,30,22,26,18,31,23,27,19,29,21,25)(33,48,40,44,36,46,38,42,34,47,39,43,35,45,37,41)(49,64,56,60,52,62,54,58,50,63,55,59,51,61,53,57)(65,80,72,76,68,78,70,74,66,79,71,75,67,77,69,73)(81,96,88,92,84,94,86,90,82,95,87,91,83,93,85,89)(97,112,104,108,100,110,102,106,98,111,103,107,99,109,101,105)(113,128,120,124,116,126,118,122,114,127,119,123,115,125,117,121)(129,144,136,140,132,142,134,138,130,143,135,139,131,141,133,137)(145,160,152,156,148,158,150,154,146,159,151,155,147,157,149,153)(161,176,168,172,164,174,166,170,162,175,167,171,163,173,165,169)(177,192,184,188,180,190,182,186,178,191,183,187,179,189,181,185)(193,208,200,204,196,206,198,202,194,207,199,203,195,205,197,201)(209,224,216,220,212,222,214,218,210,223,215,219,211,221,213,217)(225,240,232,236,228,238,230,234,226,239,231,235,227,237,229,233)(241,256,248,252,244,254,246,250,242,255,247,251,243,253,245,249), (1,8,4,6,2,7,3,5)(9,16,12,14,10,15,11,13)(17,24,20,22,18,23,19,21)(25,32,28,30,26,31,27,29)(33,40,36,38,34,39,35,37)(41,48,44,46,42,47,43,45)(49,56,52,54,50,55,51,53)(57,64,60,62,58,63,59,61)(65,72,68,70,66,71,67,69)(73,80,76,78,74,79,75,77)(81,88,84,86,82,87,83,85)(89,96,92,94,90,95,91,93)(97,104,100,102,98,103,99,101)(105,112,108,110,106,111,107,109)(113,120,116,118,114,119,115,117)(121,128,124,126,122,127,123,125)(129,136,132,134,130,135,131,133)(137,144,140,142,138,143,139,141)(145,152,148,150,146,151,147,149)(153,160,156,158,154,159,155,157)(161,168,164,166,162,167,163,165)(169,176,172,174,170,175,171,173)(177,184,180,182,178,183,179,181)(185,192,188,190,186,191,187,189)(193,200,196,198,194,199,195,197)(201,208,204,206,202,207,203,205)(209,216,212,214,210,215,211,213)(217,224,220,222,218,223,219,221)(225,232,228,230,226,231,227,229)(233,240,236,238,234,239,235,237)(241,248,244,246,242,247,243,245)(249,256,252,254,250,255,251,253), (1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,14,15)(17,20,18,19)(21,24,22,23)(25,28,26,27)(29,32,30,31)(33,36,34,35)(37,40,38,39)(41,44,42,43)(45,48,46,47)(49,52,50,51)(53,56,54,55)(57,60,58,59)(61,64,62,63)(65,68,66,67)(69,72,70,71)(73,76,74,75)(77,80,78,79)(81,84,82,83)(85,88,86,87)(89,92,90,91)(93,96,94,95)(97,100,98,99)(101,104,102,103)(105,108,106,107)(109,112,110,111)(113,116,114,115)(117,120,118,119)(121,124,122,123)(125,128,126,127)(129,132,130,131)(133,136,134,135)(137,140,138,139)(141,144,142,143)(145,148,146,147)(149,152,150,151)(153,156,154,155)(157,160,158,159)(161,164,162,163)(165,168,166,167)(169,172,170,171)(173,176,174,175)(177,180,178,179)(181,184,182,183)(185,188,186,187)(189,192,190,191)(193,196,194,195)(197,200,198,199)(201,204,202,203)(205,208,206,207)(209,212,210,211)(213,216,214,215)(217,220,218,219)(221,224,222,223)(225,228,226,227)(229,232,230,231)(233,236,234,235)(237,240,238,239)(241,244,242,243)(245,248,246,247)(249,252,250,251)(253,256,254,255), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)(223,224)(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)(239,240)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256) >;
 
Copy content gap:G := Group( (1,256,128,192,64,224,96,160,32,240,112,176,48,208,80,144,16,248,120,184,56,216,88,152,24,232,104,168,40,200,72,136,8,252,124,188,60,220,92,156,28,236,108,172,44,204,76,140,12,244,116,180,52,212,84,148,20,228,100,164,36,196,68,132,4,254,126,190,62,222,94,158,30,238,110,174,46,206,78,142,14,246,118,182,54,214,86,150,22,230,102,166,38,198,70,134,6,250,122,186,58,218,90,154,26,234,106,170,42,202,74,138,10,242,114,178,50,210,82,146,18,226,98,162,34,194,66,130,2,255,127,191,63,223,95,159,31,239,111,175,47,207,79,143,15,247,119,183,55,215,87,151,23,231,103,167,39,199,71,135,7,251,123,187,59,219,91,155,27,235,107,171,43,203,75,139,11,243,115,179,51,211,83,147,19,227,99,163,35,195,67,131,3,253,125,189,61,221,93,157,29,237,109,173,45,205,77,141,13,245,117,181,53,213,85,149,21,229,101,165,37,197,69,133,5,249,121,185,57,217,89,153,25,233,105,169,41,201,73,137,9,241,113,177,49,209,81,145,17,225,97,161,33,193,65,129), (1,128,64,96,32,112,48,80,16,120,56,88,24,104,40,72,8,124,60,92,28,108,44,76,12,116,52,84,20,100,36,68,4,126,62,94,30,110,46,78,14,118,54,86,22,102,38,70,6,122,58,90,26,106,42,74,10,114,50,82,18,98,34,66,2,127,63,95,31,111,47,79,15,119,55,87,23,103,39,71,7,123,59,91,27,107,43,75,11,115,51,83,19,99,35,67,3,125,61,93,29,109,45,77,13,117,53,85,21,101,37,69,5,121,57,89,25,105,41,73,9,113,49,81,17,97,33,65)(129,256,192,224,160,240,176,208,144,248,184,216,152,232,168,200,136,252,188,220,156,236,172,204,140,244,180,212,148,228,164,196,132,254,190,222,158,238,174,206,142,246,182,214,150,230,166,198,134,250,186,218,154,234,170,202,138,242,178,210,146,226,162,194,130,255,191,223,159,239,175,207,143,247,183,215,151,231,167,199,135,251,187,219,155,235,171,203,139,243,179,211,147,227,163,195,131,253,189,221,157,237,173,205,141,245,181,213,149,229,165,197,133,249,185,217,153,233,169,201,137,241,177,209,145,225,161,193), (1,64,32,48,16,56,24,40,8,60,28,44,12,52,20,36,4,62,30,46,14,54,22,38,6,58,26,42,10,50,18,34,2,63,31,47,15,55,23,39,7,59,27,43,11,51,19,35,3,61,29,45,13,53,21,37,5,57,25,41,9,49,17,33)(65,128,96,112,80,120,88,104,72,124,92,108,76,116,84,100,68,126,94,110,78,118,86,102,70,122,90,106,74,114,82,98,66,127,95,111,79,119,87,103,71,123,91,107,75,115,83,99,67,125,93,109,77,117,85,101,69,121,89,105,73,113,81,97)(129,192,160,176,144,184,152,168,136,188,156,172,140,180,148,164,132,190,158,174,142,182,150,166,134,186,154,170,138,178,146,162,130,191,159,175,143,183,151,167,135,187,155,171,139,179,147,163,131,189,157,173,141,181,149,165,133,185,153,169,137,177,145,161)(193,256,224,240,208,248,216,232,200,252,220,236,204,244,212,228,196,254,222,238,206,246,214,230,198,250,218,234,202,242,210,226,194,255,223,239,207,247,215,231,199,251,219,235,203,243,211,227,195,253,221,237,205,245,213,229,197,249,217,233,201,241,209,225), (1,32,16,24,8,28,12,20,4,30,14,22,6,26,10,18,2,31,15,23,7,27,11,19,3,29,13,21,5,25,9,17)(33,64,48,56,40,60,44,52,36,62,46,54,38,58,42,50,34,63,47,55,39,59,43,51,35,61,45,53,37,57,41,49)(65,96,80,88,72,92,76,84,68,94,78,86,70,90,74,82,66,95,79,87,71,91,75,83,67,93,77,85,69,89,73,81)(97,128,112,120,104,124,108,116,100,126,110,118,102,122,106,114,98,127,111,119,103,123,107,115,99,125,109,117,101,121,105,113)(129,160,144,152,136,156,140,148,132,158,142,150,134,154,138,146,130,159,143,151,135,155,139,147,131,157,141,149,133,153,137,145)(161,192,176,184,168,188,172,180,164,190,174,182,166,186,170,178,162,191,175,183,167,187,171,179,163,189,173,181,165,185,169,177)(193,224,208,216,200,220,204,212,196,222,206,214,198,218,202,210,194,223,207,215,199,219,203,211,195,221,205,213,197,217,201,209)(225,256,240,248,232,252,236,244,228,254,238,246,230,250,234,242,226,255,239,247,231,251,235,243,227,253,237,245,229,249,233,241), (1,16,8,12,4,14,6,10,2,15,7,11,3,13,5,9)(17,32,24,28,20,30,22,26,18,31,23,27,19,29,21,25)(33,48,40,44,36,46,38,42,34,47,39,43,35,45,37,41)(49,64,56,60,52,62,54,58,50,63,55,59,51,61,53,57)(65,80,72,76,68,78,70,74,66,79,71,75,67,77,69,73)(81,96,88,92,84,94,86,90,82,95,87,91,83,93,85,89)(97,112,104,108,100,110,102,106,98,111,103,107,99,109,101,105)(113,128,120,124,116,126,118,122,114,127,119,123,115,125,117,121)(129,144,136,140,132,142,134,138,130,143,135,139,131,141,133,137)(145,160,152,156,148,158,150,154,146,159,151,155,147,157,149,153)(161,176,168,172,164,174,166,170,162,175,167,171,163,173,165,169)(177,192,184,188,180,190,182,186,178,191,183,187,179,189,181,185)(193,208,200,204,196,206,198,202,194,207,199,203,195,205,197,201)(209,224,216,220,212,222,214,218,210,223,215,219,211,221,213,217)(225,240,232,236,228,238,230,234,226,239,231,235,227,237,229,233)(241,256,248,252,244,254,246,250,242,255,247,251,243,253,245,249), (1,8,4,6,2,7,3,5)(9,16,12,14,10,15,11,13)(17,24,20,22,18,23,19,21)(25,32,28,30,26,31,27,29)(33,40,36,38,34,39,35,37)(41,48,44,46,42,47,43,45)(49,56,52,54,50,55,51,53)(57,64,60,62,58,63,59,61)(65,72,68,70,66,71,67,69)(73,80,76,78,74,79,75,77)(81,88,84,86,82,87,83,85)(89,96,92,94,90,95,91,93)(97,104,100,102,98,103,99,101)(105,112,108,110,106,111,107,109)(113,120,116,118,114,119,115,117)(121,128,124,126,122,127,123,125)(129,136,132,134,130,135,131,133)(137,144,140,142,138,143,139,141)(145,152,148,150,146,151,147,149)(153,160,156,158,154,159,155,157)(161,168,164,166,162,167,163,165)(169,176,172,174,170,175,171,173)(177,184,180,182,178,183,179,181)(185,192,188,190,186,191,187,189)(193,200,196,198,194,199,195,197)(201,208,204,206,202,207,203,205)(209,216,212,214,210,215,211,213)(217,224,220,222,218,223,219,221)(225,232,228,230,226,231,227,229)(233,240,236,238,234,239,235,237)(241,248,244,246,242,247,243,245)(249,256,252,254,250,255,251,253), (1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,14,15)(17,20,18,19)(21,24,22,23)(25,28,26,27)(29,32,30,31)(33,36,34,35)(37,40,38,39)(41,44,42,43)(45,48,46,47)(49,52,50,51)(53,56,54,55)(57,60,58,59)(61,64,62,63)(65,68,66,67)(69,72,70,71)(73,76,74,75)(77,80,78,79)(81,84,82,83)(85,88,86,87)(89,92,90,91)(93,96,94,95)(97,100,98,99)(101,104,102,103)(105,108,106,107)(109,112,110,111)(113,116,114,115)(117,120,118,119)(121,124,122,123)(125,128,126,127)(129,132,130,131)(133,136,134,135)(137,140,138,139)(141,144,142,143)(145,148,146,147)(149,152,150,151)(153,156,154,155)(157,160,158,159)(161,164,162,163)(165,168,166,167)(169,172,170,171)(173,176,174,175)(177,180,178,179)(181,184,182,183)(185,188,186,187)(189,192,190,191)(193,196,194,195)(197,200,198,199)(201,204,202,203)(205,208,206,207)(209,212,210,211)(213,216,214,215)(217,220,218,219)(221,224,222,223)(225,228,226,227)(229,232,230,231)(233,236,234,235)(237,240,238,239)(241,244,242,243)(245,248,246,247)(249,252,250,251)(253,256,254,255), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)(223,224)(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)(239,240)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256) );
 
Copy content sage:G = PermutationGroup(['(1,256,128,192,64,224,96,160,32,240,112,176,48,208,80,144,16,248,120,184,56,216,88,152,24,232,104,168,40,200,72,136,8,252,124,188,60,220,92,156,28,236,108,172,44,204,76,140,12,244,116,180,52,212,84,148,20,228,100,164,36,196,68,132,4,254,126,190,62,222,94,158,30,238,110,174,46,206,78,142,14,246,118,182,54,214,86,150,22,230,102,166,38,198,70,134,6,250,122,186,58,218,90,154,26,234,106,170,42,202,74,138,10,242,114,178,50,210,82,146,18,226,98,162,34,194,66,130,2,255,127,191,63,223,95,159,31,239,111,175,47,207,79,143,15,247,119,183,55,215,87,151,23,231,103,167,39,199,71,135,7,251,123,187,59,219,91,155,27,235,107,171,43,203,75,139,11,243,115,179,51,211,83,147,19,227,99,163,35,195,67,131,3,253,125,189,61,221,93,157,29,237,109,173,45,205,77,141,13,245,117,181,53,213,85,149,21,229,101,165,37,197,69,133,5,249,121,185,57,217,89,153,25,233,105,169,41,201,73,137,9,241,113,177,49,209,81,145,17,225,97,161,33,193,65,129)', '(1,128,64,96,32,112,48,80,16,120,56,88,24,104,40,72,8,124,60,92,28,108,44,76,12,116,52,84,20,100,36,68,4,126,62,94,30,110,46,78,14,118,54,86,22,102,38,70,6,122,58,90,26,106,42,74,10,114,50,82,18,98,34,66,2,127,63,95,31,111,47,79,15,119,55,87,23,103,39,71,7,123,59,91,27,107,43,75,11,115,51,83,19,99,35,67,3,125,61,93,29,109,45,77,13,117,53,85,21,101,37,69,5,121,57,89,25,105,41,73,9,113,49,81,17,97,33,65)(129,256,192,224,160,240,176,208,144,248,184,216,152,232,168,200,136,252,188,220,156,236,172,204,140,244,180,212,148,228,164,196,132,254,190,222,158,238,174,206,142,246,182,214,150,230,166,198,134,250,186,218,154,234,170,202,138,242,178,210,146,226,162,194,130,255,191,223,159,239,175,207,143,247,183,215,151,231,167,199,135,251,187,219,155,235,171,203,139,243,179,211,147,227,163,195,131,253,189,221,157,237,173,205,141,245,181,213,149,229,165,197,133,249,185,217,153,233,169,201,137,241,177,209,145,225,161,193)', '(1,64,32,48,16,56,24,40,8,60,28,44,12,52,20,36,4,62,30,46,14,54,22,38,6,58,26,42,10,50,18,34,2,63,31,47,15,55,23,39,7,59,27,43,11,51,19,35,3,61,29,45,13,53,21,37,5,57,25,41,9,49,17,33)(65,128,96,112,80,120,88,104,72,124,92,108,76,116,84,100,68,126,94,110,78,118,86,102,70,122,90,106,74,114,82,98,66,127,95,111,79,119,87,103,71,123,91,107,75,115,83,99,67,125,93,109,77,117,85,101,69,121,89,105,73,113,81,97)(129,192,160,176,144,184,152,168,136,188,156,172,140,180,148,164,132,190,158,174,142,182,150,166,134,186,154,170,138,178,146,162,130,191,159,175,143,183,151,167,135,187,155,171,139,179,147,163,131,189,157,173,141,181,149,165,133,185,153,169,137,177,145,161)(193,256,224,240,208,248,216,232,200,252,220,236,204,244,212,228,196,254,222,238,206,246,214,230,198,250,218,234,202,242,210,226,194,255,223,239,207,247,215,231,199,251,219,235,203,243,211,227,195,253,221,237,205,245,213,229,197,249,217,233,201,241,209,225)', '(1,32,16,24,8,28,12,20,4,30,14,22,6,26,10,18,2,31,15,23,7,27,11,19,3,29,13,21,5,25,9,17)(33,64,48,56,40,60,44,52,36,62,46,54,38,58,42,50,34,63,47,55,39,59,43,51,35,61,45,53,37,57,41,49)(65,96,80,88,72,92,76,84,68,94,78,86,70,90,74,82,66,95,79,87,71,91,75,83,67,93,77,85,69,89,73,81)(97,128,112,120,104,124,108,116,100,126,110,118,102,122,106,114,98,127,111,119,103,123,107,115,99,125,109,117,101,121,105,113)(129,160,144,152,136,156,140,148,132,158,142,150,134,154,138,146,130,159,143,151,135,155,139,147,131,157,141,149,133,153,137,145)(161,192,176,184,168,188,172,180,164,190,174,182,166,186,170,178,162,191,175,183,167,187,171,179,163,189,173,181,165,185,169,177)(193,224,208,216,200,220,204,212,196,222,206,214,198,218,202,210,194,223,207,215,199,219,203,211,195,221,205,213,197,217,201,209)(225,256,240,248,232,252,236,244,228,254,238,246,230,250,234,242,226,255,239,247,231,251,235,243,227,253,237,245,229,249,233,241)', '(1,16,8,12,4,14,6,10,2,15,7,11,3,13,5,9)(17,32,24,28,20,30,22,26,18,31,23,27,19,29,21,25)(33,48,40,44,36,46,38,42,34,47,39,43,35,45,37,41)(49,64,56,60,52,62,54,58,50,63,55,59,51,61,53,57)(65,80,72,76,68,78,70,74,66,79,71,75,67,77,69,73)(81,96,88,92,84,94,86,90,82,95,87,91,83,93,85,89)(97,112,104,108,100,110,102,106,98,111,103,107,99,109,101,105)(113,128,120,124,116,126,118,122,114,127,119,123,115,125,117,121)(129,144,136,140,132,142,134,138,130,143,135,139,131,141,133,137)(145,160,152,156,148,158,150,154,146,159,151,155,147,157,149,153)(161,176,168,172,164,174,166,170,162,175,167,171,163,173,165,169)(177,192,184,188,180,190,182,186,178,191,183,187,179,189,181,185)(193,208,200,204,196,206,198,202,194,207,199,203,195,205,197,201)(209,224,216,220,212,222,214,218,210,223,215,219,211,221,213,217)(225,240,232,236,228,238,230,234,226,239,231,235,227,237,229,233)(241,256,248,252,244,254,246,250,242,255,247,251,243,253,245,249)', '(1,8,4,6,2,7,3,5)(9,16,12,14,10,15,11,13)(17,24,20,22,18,23,19,21)(25,32,28,30,26,31,27,29)(33,40,36,38,34,39,35,37)(41,48,44,46,42,47,43,45)(49,56,52,54,50,55,51,53)(57,64,60,62,58,63,59,61)(65,72,68,70,66,71,67,69)(73,80,76,78,74,79,75,77)(81,88,84,86,82,87,83,85)(89,96,92,94,90,95,91,93)(97,104,100,102,98,103,99,101)(105,112,108,110,106,111,107,109)(113,120,116,118,114,119,115,117)(121,128,124,126,122,127,123,125)(129,136,132,134,130,135,131,133)(137,144,140,142,138,143,139,141)(145,152,148,150,146,151,147,149)(153,160,156,158,154,159,155,157)(161,168,164,166,162,167,163,165)(169,176,172,174,170,175,171,173)(177,184,180,182,178,183,179,181)(185,192,188,190,186,191,187,189)(193,200,196,198,194,199,195,197)(201,208,204,206,202,207,203,205)(209,216,212,214,210,215,211,213)(217,224,220,222,218,223,219,221)(225,232,228,230,226,231,227,229)(233,240,236,238,234,239,235,237)(241,248,244,246,242,247,243,245)(249,256,252,254,250,255,251,253)', '(1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,14,15)(17,20,18,19)(21,24,22,23)(25,28,26,27)(29,32,30,31)(33,36,34,35)(37,40,38,39)(41,44,42,43)(45,48,46,47)(49,52,50,51)(53,56,54,55)(57,60,58,59)(61,64,62,63)(65,68,66,67)(69,72,70,71)(73,76,74,75)(77,80,78,79)(81,84,82,83)(85,88,86,87)(89,92,90,91)(93,96,94,95)(97,100,98,99)(101,104,102,103)(105,108,106,107)(109,112,110,111)(113,116,114,115)(117,120,118,119)(121,124,122,123)(125,128,126,127)(129,132,130,131)(133,136,134,135)(137,140,138,139)(141,144,142,143)(145,148,146,147)(149,152,150,151)(153,156,154,155)(157,160,158,159)(161,164,162,163)(165,168,166,167)(169,172,170,171)(173,176,174,175)(177,180,178,179)(181,184,182,183)(185,188,186,187)(189,192,190,191)(193,196,194,195)(197,200,198,199)(201,204,202,203)(205,208,206,207)(209,212,210,211)(213,216,214,215)(217,220,218,219)(221,224,222,223)(225,228,226,227)(229,232,230,231)(233,236,234,235)(237,240,238,239)(241,244,242,243)(245,248,246,247)(249,252,250,251)(253,256,254,255)', '(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)(223,224)(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)(239,240)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256)'])
 
Matrix group:$\left\langle \left(\begin{array}{rr} 72 & 76 \\ 110 & 72 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{127})$
Copy content comment:Define the group as a matrix group with coefficients in GLFp
 
Copy content magma:G := MatrixGroup< 2, GF(127) | [[72, 76, 110, 72]] >;
 
Copy content gap:G := Group([[[ Z(127)^92, Z(127)^102 ], [ Z(127)^101, Z(127)^92 ]]]);
 
Copy content sage:MS = MatrixSpace(GF(127), 2, 2) G = MatrixGroup([MS([[72, 76], [110, 72]])])
 
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: not isomorphic to a non-trivial semidirect product
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $C_{64}$ . $C_4$ $C_{32}$ . $C_8$ $C_8$ . $C_{32}$ $C_4$ . $C_{64}$ all 7
Aut. group: $\Aut(C_{257})$ $\Aut(C_{514})$

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Primary decomposition: $C_{256}$
Copy content comment:The primary decomposition of the group
 
Copy content magma:PrimaryInvariants(G);
 
Copy content gap:AbelianInvariants(G);
 
Copy content sage_gap:G.AbelianInvariants()
 
Schur multiplier: $C_1$
Copy content comment:The Schur multiplier of the group
 
Copy content gap:AbelianInvariantsMultiplier(G);
 
Copy content sage:G.homology(2)
 
Copy content sage_gap:G.AbelianInvariantsMultiplier()
 
Commutator length: $0$
Copy content comment:The commutator length of the group
 
Copy content gap:CommutatorLength(G);
 
Copy content sage_gap:G.CommutatorLength()
 

Subgroups

Copy content comment:List of subgroups of the group
 
Copy content magma:Subgroups(G);
 
Copy content gap:AllSubgroups(G);
 
Copy content sage:G.subgroups()
 
Copy content sage_gap:G.AllSubgroups()
 

There are 9 subgroups, all normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_{256}$ $G/Z \simeq$ $C_1$
Copy content comment:Center of the group
 
Copy content magma:Center(G);
 
Copy content gap:Center(G);
 
Copy content sage:G.center()
 
Copy content sage_gap:G.Center()
 
Commutator: $G' \simeq$ $C_1$ $G/G' \simeq$ $C_{256}$
Copy content comment:Commutator subgroup of the group G
 
Copy content magma:CommutatorSubgroup(G);
 
Copy content gap:DerivedSubgroup(G);
 
Copy content sage:G.commutator()
 
Copy content sage_gap:G.DerivedSubgroup()
 
Frattini: $\Phi \simeq$ $C_{128}$ $G/\Phi \simeq$ $C_2$
Copy content comment:Frattini subgroup of the group G
 
Copy content magma:FrattiniSubgroup(G);
 
Copy content gap:FrattiniSubgroup(G);
 
Copy content sage:G.frattini_subgroup()
 
Copy content sage_gap:G.FrattiniSubgroup()
 
Fitting: $\operatorname{Fit} \simeq$ $C_{256}$ $G/\operatorname{Fit} \simeq$ $C_1$
Copy content comment:Fitting subgroup of the group G
 
Copy content magma:FittingSubgroup(G);
 
Copy content gap:FittingSubgroup(G);
 
Copy content sage:G.fitting_subgroup()
 
Copy content sage_gap:G.FittingSubgroup()
 
Radical: $R \simeq$ $C_{256}$ $G/R \simeq$ $C_1$
Copy content comment:Radical of the group G
 
Copy content magma:Radical(G);
 
Copy content gap:SolvableRadical(G);
 
Copy content sage_gap:G.SolvableRadical()
 
Socle: $\operatorname{soc} \simeq$ $C_2$ $G/\operatorname{soc} \simeq$ $C_{128}$
Copy content comment:Socle of the group G
 
Copy content magma:Socle(G);
 
Copy content gap:Socle(G);
 
Copy content sage:G.socle()
 
Copy content sage_gap:G.Socle()
 
2-Sylow subgroup: $P_{ 2 } \simeq$ $C_{256}$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $C_{256}$ $\rhd$ $C_1$
Copy content comment:Derived series of the group GF
 
Copy content magma:DerivedSeries(G);
 
Copy content gap:DerivedSeriesOfGroup(G);
 
Copy content sage:G.derived_series()
 
Copy content sage_gap:G.DerivedSeriesOfGroup()
 
Chief series $C_{256}$ $\rhd$ $C_{128}$ $\rhd$ $C_{64}$ $\rhd$ $C_{32}$ $\rhd$ $C_{16}$ $\rhd$ $C_8$ $\rhd$ $C_4$ $\rhd$ $C_2$ $\rhd$ $C_1$
Copy content comment:Chief series of the group G
 
Copy content magma:ChiefSeries(G);
 
Copy content gap:ChiefSeries(G);
 
Copy content sage_gap:G.ChiefSeries()
 
Lower central series $C_{256}$ $\rhd$ $C_1$
Copy content comment:The lower central series of the group G
 
Copy content magma:LowerCentralSeries(G);
 
Copy content gap:LowerCentralSeriesOfGroup(G);
 
Copy content sage:G.lower_central_series()
 
Copy content sage_gap:G.LowerCentralSeriesOfGroup()
 
Upper central series $C_1$ $\lhd$ $C_{256}$
Copy content comment:The upper central series of the group G
 
Copy content magma:UpperCentralSeries(G);
 
Copy content gap:UpperCentralSeriesOfGroup(G);
 
Copy content sage:G.upper_central_series()
 
Copy content sage_gap:G.UpperCentralSeriesOfGroup()
 

Supergroups

This group is a maximal subgroup of 21 larger groups in the database.

This group is a maximal quotient of 17 larger groups in the database.

Character theory

Copy content comment:Character table
 
Copy content magma:CharacterTable(G); // Output not guaranteed to exactly match the LMFDB table
 
Copy content gap:CharacterTable(G); # Output not guaranteed to exactly match the LMFDB table
 
Copy content sage:G.character_table() # Output not guaranteed to exactly match the LMFDB table
 
Copy content sage_gap:G.CharacterTable() # Output not guaranteed to exactly match the LMFDB table
 

Complex character table

See the $256 \times 256$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

1A 2A 4A 8A 16A 32A 64A 128A 256A
Size 1 1 2 4 8 16 32 64 128
2 P 1A 1A 2A 4A 8A 16A 32A 64A 128A
256.1.1a 1 1 1 1 1 1 1 1 1
256.1.1b 1 1 1 1 1 1 1 1 −1
256.1.1c 2 2 2 2 2 2 2 −2 0
256.1.1d 4 4 4 4 4 4 −4 0 0
256.1.1e 8 8 8 8 8 −8 0 0 0
256.1.1f 16 16 16 16 −16 0 0 0 0
256.1.1g 32 32 32 −32 0 0 0 0 0
256.1.1h 64 64 −64 0 0 0 0 0 0
256.1.1i 128 −128 0 0 0 0 0 0 0