#7624866

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 ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1301 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:

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

Polycyclic presentation:

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

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, 7532062);