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Properties

Label 512.7532443
Order \( 2^{9} \)
Exponent \( 2^{2} \)
Nilpotent yes
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2^{7} \)
$\card{Z(G)}$ \( 2^{6} \)
$\card{\Aut(G)}$ \( 2^{28} \cdot 3^{2} \)
$\card{\mathrm{Out}(G)}$ \( 2^{25} \cdot 3^{2} \)
Perm deg. not computed
Trans deg. $128$
Rank $6$

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Show commands: Gap / Magma / SageMath (using Gap)

Copy content magma:G := SmallGroup(512, 7532443);
 
Copy content gap:G := SmallGroup(512, 7532443);
 
Copy content sage:G = libgap.SmallGroup(512, 7532443)
 
Copy content comment:Define the group with the given generators and relations
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(1277558994098656108461082838537,512)'); a = G.1; b = G.3; c = G.4; d = G.6; e = G.7; f = G.8; g = G.9;
 

Group information

Description:$C_2^3.C_2^6$
Order: \(512\)\(\medspace = 2^{9} \)
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:G.Order()
 
Exponent: \(4\)\(\medspace = 2^{2} \)
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:G.Exponent()
 
Automorphism group:Group of order \(2415919104\)\(\medspace = 2^{28} \cdot 3^{2} \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage:G.AutomorphismGroup()
 
Composition factors:$C_2$ x 9
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:G.CompositionSeries()
 
Nilpotency class:$2$
Copy content comment:Nilpotency class of the group
 
Copy content magma:NilpotencyClass(G);
 
Copy content gap:NilpotencyClassOfGroup(G);
 
Copy content sage:G.NilpotencyClassOfGroup()
 
Derived length:$2$
Copy content comment:Derived length of the group
 
Copy content magma:DerivedLength(G);
 
Copy content gap:DerivedLength(G);
 
Copy content sage:G.DerivedLength()
 

This group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.

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: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: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: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: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: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: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
Elements 1 159 352 512
Conjugacy classes   1 87 136 224
Divisions 1 87 80 168
Autjugacy classes 1 6 5 12

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:G.CharacterDegrees()
 

Dimension 1 2 4
Irr. complex chars.   128 96 0 224
Irr. rational chars. 64 80 24 168

Minimal presentations

Permutation degree:not computed
Transitive degree:$128$
Rank: $6$
Inequivalent generating 6-tuples: $2187360$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible none none none
Arbitrary not computed not computed not computed

Constructions

Show commands: Gap / Magma / SageMath (using Gap)


Presentation: ${\langle a, b, c, d, e, f, g \mid a^{4}=b^{2}=c^{4}=d^{2}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([9, 2, 2, 2, 2, 2, 2, 2, 2, 2, 18, 542, 867, 237, 102, 25928, 3275]); a,b,c,d,e,f,g := Explode([G.1, G.3, G.4, G.6, G.7, G.8, G.9]); AssignNames(~G, ["a", "a2", "b", "c", "c2", "d", "e", "f", "g"]);
 
Copy content gap:G := PcGroupCode(1277558994098656108461082838537,512); a := G.1; b := G.3; c := G.4; d := G.6; e := G.7; f := G.8; g := G.9;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(1277558994098656108461082838537,512)'); a = G.1; b = G.3; c = G.4; d = G.6; e = G.7; f = G.8; g = G.9;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(1277558994098656108461082838537,512)'); a = G.1; b = G.3; c = G.4; d = G.6; e = G.7; f = G.8; g = G.9;
 
Direct product: $C_2$ ${}^2$ $\, \times\, $ $C_4$ $\, \times\, $ $(C_2^2\wr C_2)$
Semidirect product: $(C_2^5.D_4)$ $\,\rtimes\,$ $C_2$ $(C_2^5.D_4)$ $\,\rtimes\,$ $C_2$ $(C_2^4\times C_4)$ $\,\rtimes\,$ $D_4$ $C_2^4$ $\,\rtimes\,$ $(C_4\times D_4)$ all 50
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $C_2^7$ . $C_2^2$ $C_2^6$ . $C_2^3$ (3) $C_2^5$ . $C_2^4$ (4) $C_2^4$ . $C_2^5$ (3) all 42
Aut. group: $\Aut(D_4\times C_{16})$ $\Aut(C_{16}:D_4)$ $\Aut(C_{16}:Q_8)$ $\Aut(D_4\times C_{20})$ all 12

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

Homology

Abelianization: $C_{2}^{5} \times C_{4} $
Copy content comment:The abelianization of the group
 
Copy content magma:quo< G | CommutatorSubgroup(G) >;
 
Copy content gap:FactorGroup(G, DerivedSubgroup(G));
 
Copy content sage:G.quotient(G.commutator())
 
Schur multiplier: $C_{2}^{16}$
Copy content comment:The Schur multiplier of the group
 
Copy content gap:AbelianInvariantsMultiplier(G);
 
Copy content sage:G.homology(2)
 
Copy content sage:G.AbelianInvariantsMultiplier()
 
Commutator length: $1$
Copy content comment:The commutator length of the group
 
Copy content gap:CommutatorLength(G);
 
Copy content sage: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:G.AllSubgroups()
 

There are 76674 subgroups in 41106 conjugacy classes, 7346 normal (14 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_2^4\times C_4$ $G/Z \simeq$ $C_2^3$
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:G.Center()
 
Commutator: $G' \simeq$ $C_2^2$ $G/G' \simeq$ $C_2^5\times C_4$
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:G.DerivedSubgroup()
 
Frattini: $\Phi \simeq$ $C_2^3$ $G/\Phi \simeq$ $C_2^6$
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:G.FrattiniSubgroup()
 
Fitting: $\operatorname{Fit} \simeq$ $C_2^3.C_2^6$ $G/\operatorname{Fit} \simeq$ $C_1$
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:G.FittingSubgroup()
 
Radical: $R \simeq$ $C_2^3.C_2^6$ $G/R \simeq$ $C_1$
Copy content comment:Radical of the group G
 
Copy content magma:Radical(G);
 
Copy content gap:SolvableRadical(G);
 
Copy content sage:G.SolvableRadical()
 
Socle: $\operatorname{soc} \simeq$ $C_2^5$ $G/\operatorname{soc} \simeq$ $C_2^4$
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:G.Socle()
 
2-Sylow subgroup: $P_{ 2 } \simeq$ $C_2^3.C_2^6$

Subgroup diagram and profile

Series

Derived series $C_2^3.C_2^6$ $\rhd$ $C_2^2$ $\rhd$ $C_1$
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:G.DerivedSeriesOfGroup()
 
Chief series $C_2^3.C_2^6$ $\rhd$ $C_2^5:D_4$ $\rhd$ $C_2^5:C_4$ $\rhd$ $C_2^6$ $\rhd$ $C_2^5$ $\rhd$ $C_2^4$ $\rhd$ $C_2^3$ $\rhd$ $C_2^2$ $\rhd$ $C_2$ $\rhd$ $C_1$
Copy content comment:Chief series of the group G
 
Copy content magma:ChiefSeries(G);
 
Copy content gap:ChiefSeries(G);
 
Copy content sage:G.ChiefSeries()
 
Lower central series $C_2^3.C_2^6$ $\rhd$ $C_2^2$ $\rhd$ $C_1$
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:G.LowerCentralSeriesOfGroup()
 
Upper central series $C_1$ $\lhd$ $C_2^4\times C_4$ $\lhd$ $C_2^3.C_2^6$
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:G.UpperCentralSeriesOfGroup()
 

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

This group is a maximal quotient of 1 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:G.CharacterTable() # Output not guaranteed to exactly match the LMFDB table
 

Complex character table

See the $224 \times 224$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $168 \times 168$ rational character table (warning: may be slow to load).