#7624096

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623613
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

1292 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)(6,8)(9,10,13,17)(11,20,22,38)(12,18,15,24)(14,27,26,37)(16,31,32,23)(19,25,35,29)(21,39,30,36)(28,33,40,34);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,24)(13,17)(14,19)(15,18)(20,23)(21,34)(22,32)(25,37)(26,35)(27,29)(28,36)(30,33)(31,38)(39,40);(9,11,13,22)(10,16,17,32)(12,21,15,30)(14,28,26,40)(18,33,24,34)(19,36,35,39)(20,37,38,27)(23,25,31,29);(2,5)(6,8)(10,17)(16,32)(18,24)(19,35)(20,38)(27,37)(33,34)(36,39);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,18,17,24)(11,21,22,30)(14,25,26,29)(16,33,32,34)(19,27,35,37)(20,39,38,36)(23,28,31,40)

Polycyclic presentation:

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

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, 7531292);