#72321374

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#442
commutator	C₈	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
socle	C₂³	C₂×#442
prime radical	C₂²×#135	C₂×C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1833 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,3,4)(5,6,8,10,9,11,12,7)(13,14)(15,23,24,39)(16,27)(17,31)(18,20)(19,26,35,41)(21,30)(22,33)(25,37,32,40)(28,38,42,34)(29,43)(36,44);(2,4)(13,15,18,24)(14,19,20,35)(16,28,29,42)(17,32,33,25)(21,37,36,40)(22,38,31,34)(23,27,39,43)(26,30,41,44);(14,20)(15,24)(21,36)(22,31)(25,32)(26,41)(27,43)(28,42);(5,7,12,11,9,10,8,6)(15,25,24,32)(16,29)(19,34,35,38)(21,36)(23,37,39,40)(26,28,41,42)(27,43)(30,44)

Polycyclic presentation:

2^10|a^2=e,b^2=j,d^2=efgij,e^2=h,f^2=i,g^2=i,i^2=j,b^a=bf,c^a=cj,c^b=cj,d^b=dij,e^b=ej,f^a=fi,f^b=fij,f^d=fj,g^a=gi,g^b=gij,g^d=gj,i^a=ij,i^b=ij

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321461, #72321463, #72321904
maximal normal subgroup of	#72321461, #72321463, #72321904
maximal quotient of	#72322586

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 4, F.5*F.6*F.7*F.9*F.10);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);