#7624906

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624888
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 ◁ #525 ◁ #2929 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1299 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)(3,7)(4,6)(5,8)(9,10,15,20)(11,16,21,32)(12,23,25,39)(13,28,30,40)(14,31,29,19)(17,33,34,26)(18,35,36,24)(22,38,37,27);(1,3)(2,5)(4,8)(6,7)(9,11,15,21)(10,16,20,32)(12,18,25,36)(13,22,30,37)(14,17,29,34)(19,26,31,33)(23,35,39,24)(27,28,38,40);(9,12,24,32,14,27,30,33,15,25,35,16,29,38,13,26)(10,17,28,37,19,21,36,39,20,34,40,22,31,11,18,23);(3,8)(5,7);(2,6)(5,7)(11,21)(12,25)(16,32)(17,34)(22,37)(23,39)(26,33)(27,38)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532102);