#7625002

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624842
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 ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1391 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:

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

Polycyclic presentation:

2^9|c^2=gh,e^2=fi,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,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532198);