#7624996

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624824
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

1398 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)(13,27)(14,26)(15,19)(16,23)(18,28)(21,33)(22,34)(24,40)(25,36)(29,30)(31,37)(32,39)(35,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,26)(13,18)(14,17)(15,19)(20,23)(21,30)(22,35)(24,31)(25,32)(27,28)(29,33)(34,38)(36,39)(37,40);(9,11,21,37,14,24,13,23,15,25,28,38,12,22,29,39)(10,16,30,40,17,31,18,20,19,32,27,34,26,35,33,36);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,12,15,14)(10,17,19,26)(11,22,25,24)(13,21,29,28)(16,31,32,35)(18,27,33,30)(20,34,36,40)(23,37,39,38)

Polycyclic presentation:

2^9|c^2=ghi,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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532192);