#7624227

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
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 ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1511 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=i,d^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=dfi,e^b=ei,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531423);