#7623266

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622982
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

1372 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)(10,15)(12,23)(13,24)(16,26)(17,25)(18,27)(19,31)(21,36)(22,35)(28,29)(30,37)(32,34)(33,40)(38,39);(2,5)(6,8)(10,16)(15,26)(18,30)(19,28)(27,37)(29,31)(33,38)(39,40);(9,10,17,31,13,19,12,15,14,16,23,29,24,28,25,26)(11,18,32,40,22,33,21,27,20,30,36,39,35,38,34,37);(1,3,4,7)(2,6,5,8)(9,11)(10,18)(12,21)(13,22)(14,20)(15,27)(16,30)(17,32)(19,33)(23,36)(24,35)(25,34)(26,37)(28,38)(29,39)(31,40);(11,20)(18,30)(21,34)(22,35)(27,37)(32,36)(33,38)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530462);