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Properties

Label 512.10494213
Order \( 2^{9} \)
Exponent \( 2 \)
Abelian yes
$\card{\Aut(G)}$ \( 2^{36} \cdot 3^{5} \cdot 5^{2} \cdot 7^{3} \cdot 17 \cdot 31 \cdot 73 \cdot 127 \)
Perm deg. $18$
Trans deg. $512$
Rank $9$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := SmallGroup(512, 10494213);
 
Copy content gap:G := SmallGroup(512, 10494213);
 
Copy content sage_gap:G = libgap.SmallGroup(512, 10494213)
 
Copy content comment:Define the group as a permutation group
 
Copy content sage:G = PermutationGroup(['(1,2)', '(3,4)', '(5,6)', '(7,8)', '(9,10)', '(11,12)', '(13,14)', '(15,16)', '(17,18)'])
 

Group information

Description:$C_2^9$
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_gap:G.Order()
 
Exponent: \(2\)
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_gap:G.Exponent()
 
Automorphism group:$\GL(9,2)$, of order \(699\!\cdots\!200\)\(\medspace = 2^{36} \cdot 3^{5} \cdot 5^{2} \cdot 7^{3} \cdot 17 \cdot 31 \cdot 73 \cdot 127 \)
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 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_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 abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational.

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
Elements 1 511 512
Conjugacy classes   1 511 512
Divisions 1 511 512
Autjugacy classes 1 1 2

Minimal presentations

Permutation degree:$18$
Transitive degree:$512$
Rank: $9$
Inequivalent generating 9-tuples: $1$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible none not computed none
Arbitrary 9 not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


Presentation:Abelian group $\langle a, b, c, d, e, f, g, h, i \mid a^{2}=b^{2}=c^{2}=d^{2}=e^{2}=f^{2}=g^{2}=h^{2}=i^{2}=1 \rangle$ 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]); a,b,c,d,e,f,g,h,i := Explode([G.1, G.2, G.3, G.4, G.5, G.6, G.7, G.8, G.9]); AssignNames(~G, ["a", "b", "c", "d", "e", "f", "g", "h", "i"]);
 
Copy content gap:G := PcGroupCode(0,512); a := G.1; b := G.2; c := G.3; d := G.4; e := G.5; f := G.6; g := G.7; h := G.8; i := 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(0,512)'); a = G.1; b = G.2; c = G.3; d = G.4; e = G.5; f = G.6; g = G.7; h = G.8; i = G.9;
 
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(0,512)'); a = G.1; b = G.2; c = G.3; d = G.4; e = G.5; f = G.6; g = G.7; h = G.8; i = G.9;
 
Permutation group:Degree $18$ $\langle(1,2), (3,4), (5,6), (7,8), (9,10), (11,12), (13,14), (15,16), (17,18)\rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 18 | (1,2), (3,4), (5,6), (7,8), (9,10), (11,12), (13,14), (15,16), (17,18) >;
 
Copy content gap:G := Group( (1,2), (3,4), (5,6), (7,8), (9,10), (11,12), (13,14), (15,16), (17,18) );
 
Copy content sage:G = PermutationGroup(['(1,2)', '(3,4)', '(5,6)', '(7,8)', '(9,10)', '(11,12)', '(13,14)', '(15,16)', '(17,18)'])
 
Direct product: $C_2$ ${}^9$
Semidirect product: not isomorphic to a non-trivial semidirect product
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Aut. group: $\Aut(C_2^3.C_2^3)$ $\Aut(C_2^3.C_2^3)$

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

Homology

Primary decomposition: $C_{2}^{9}$
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: $C_{2}^{36}$
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 8283458 subgroups, all normal (2 characteristic).

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

Special subgroups

Center: $Z \simeq$ $C_2^9$ $G/Z \simeq$ $C_1$
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: $G' \simeq$ $C_1$ $G/G' \simeq$ $C_2^9$
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: $\Phi \simeq$ $C_1$ $G/\Phi \simeq$ $C_2^9$
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: $\operatorname{Fit} \simeq$ $C_2^9$ $G/\operatorname{Fit} \simeq$ $C_1$
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: $R \simeq$ $C_2^9$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_2^9$ $G/\operatorname{soc} \simeq$ $C_1$
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: $P_{ 2 } \simeq$ $C_2^9$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $C_2^9$ $\rhd$ $C_1$
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 $C_2^9$ $\rhd$ $C_2^8$ $\rhd$ $C_2^7$ $\rhd$ $C_2^6$ $\rhd$ $C_2^5$ $\rhd$ $C_2^4$ $\rhd$ $C_2^3$ $\rhd$ $C_2^2$ $\rhd$ $C_2$ $\rhd$ $C_1$
Copy content comment:Chief series of the group G
 
Copy content magma:ChiefSeries(G);
 
Copy content gap:ChiefSeries(G);
 
Copy content sage_gap:G.ChiefSeries()
 
Lower central series $C_2^9$ $\rhd$ $C_1$
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 $C_1$ $\lhd$ $C_2^9$
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 10 larger groups in the database.

This group is a maximal quotient of 3 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

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

The $512 \times 512$ rational character table is not available for this group.