#72286970

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₈² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×#322
commutator	C₂×C₈	C₂×C₈²
Frattini, cosocle	C₂²×#346	C₂³
socle	C₂³	C₂×#1313
prime radical	C₂⁵	#347
LCS	G ▷ C₂×C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×#346 ◁ G
LECS	G ▷ C₂²×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2872 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,4)(5,6)(7,8);(1,2)(3,4)(10,16,18,20,21,33,37,22)(11,23,25,28)(12,13)(17,27,35,26)(19,34,24,40,39,38,36,32)(30,31);(2,4)(6,8)(9,10,17,34,31,33,28,19,29,37,26,40,13,20,11,24,15,21,35,38,30,16,23,39,14,18,27,32,12,22,25,36)

Polycyclic presentation:

2^11|b^2=d,c^2=dfijk,d^2=h,f^2=i,g^2=j,i^2=j,j^2=k,c^a=ce,c^b=cegjk,d^c=djk,f^b=fjk,f^c=fjk,f^d=fk,g^b=gk,g^c=gk,h^c=hk,i^b=ik,i^c=ik

Conjugacy classes & character table

272 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286981, #72287029, #72287634
maximal normal subgroup of	#72286981, #72287029, #72287634

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.4*F.6*F.9*F.10*F.11);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.7*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);