#7624879

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624876
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

1313 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)(11,16,14,29)(12,22,24,36)(13,27,28,40)(17,31,32,25)(18,21,33,23)(20,34,26,39)(30,38,37,35);(1,3)(2,5)(4,8)(6,7)(9,11,15,14)(10,16,19,29)(12,20,24,26)(13,21,28,23)(17,30,32,37)(18,27,33,40)(22,34,36,39)(25,35,31,38);(9,12,23,38,14,26,28,31,15,24,21,35,11,20,13,25)(10,17,27,34,16,30,33,36,19,32,40,39,29,37,18,22);(3,8)(5,7);(2,6)(5,7)(12,24)(17,32)(20,26)(22,36)(25,31)(30,37)(34,39)(35,38)

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^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, 7532075);