#7624810

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624802
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×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1324 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)(9,10,15,19)(11,20,25,36)(12,17,14,29)(13,27,28,40)(16,30,32,23)(18,26,33,21)(22,34,24,39)(31,37,35,38);(1,3)(2,5)(4,8)(6,7);(9,11,21,37,14,24,28,30,15,25,26,38,12,22,13,23)(10,16,27,34,17,31,33,36,19,32,40,39,29,35,18,20);(3,8)(5,7);(2,6)(5,7)(9,12,15,14)(10,17,19,29)(11,22,25,24)(13,26,28,21)(16,31,32,35)(18,27,33,40)(20,34,36,39)(23,38,30,37)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532006);