#7624036

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623604
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1334 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321856, #72343987
maximal normal subgroup of	#72321856, #72343987

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531232);