#72343571

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#442
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#13325	C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #8260 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

958 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=e,c^2=fh,d^2=f,e^2=h,g^2=i,i^2=j,b^a=bi,c^b=cf,d^a=dg,d^b=dg,d^c=dh,e^d=ej,g^a=gi,g^b=gij,g^d=gij,i^a=ij,i^b=ij,i^d=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343582, #72343627, #72343776
maximal normal subgroup of	#72343582, #72343627, #72343776

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