#7624106

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623626
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

1320 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)(9,10,14,17)(11,15,20,30)(12,16,25,31)(13,26,24,19)(18,33,32,23)(21,27,34,39)(22,37,35,29)(28,40,38,36);(1,2)(3,6)(4,5)(7,8)(9,11,14,20)(10,15,17,30)(12,21,25,34)(13,22,24,35)(16,27,31,39)(18,28,32,38)(19,29,26,37)(23,36,33,40);(10,16)(12,23)(13,24)(15,27)(17,31)(18,19)(21,36)(22,35)(25,33)(26,32)(28,29)(30,39)(34,40)(37,38);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,17)(11,20)(16,31)(18,32)(19,26)(21,34)(22,35)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322867
maximal normal subgroup of	#72322867

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531302);