#7624208

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624170
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

1509 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)(10,17)(11,21)(13,16)(15,28)(18,33)(20,25)(22,37)(23,39)(24,38)(26,27)(29,30)(31,35)(32,36)(34,40);(1,3)(2,5)(4,8)(6,7)(9,10,14,17)(11,18,24,31)(12,16,19,13)(15,20,27,32)(21,35,38,33)(22,30,34,23)(25,28,36,26)(29,37,39,40);(9,11)(10,18)(12,22)(13,23)(14,24)(15,29)(16,30)(17,31)(19,34)(20,37)(21,28)(25,33)(26,38)(27,39)(32,40)(35,36);(3,8)(5,7)(9,12,14,19)(10,13,17,16)(11,22,24,34)(15,25,27,36)(18,23,31,30)(20,26,32,28)(21,37,38,40)(29,33,39,35);(2,6)(5,7)(9,13,14,16)(10,19,17,12)(11,23,24,30)(15,26,27,28)(18,34,31,22)(20,36,32,25)(21,29,38,39)(33,37,35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531404);