#7623259

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622976
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 ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1406 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313603, #72313605, #72313684, …
maximal normal subgroup of	#72313603, #72313605, #72313684, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530455);