#6631416

order	512
nilpotency class	3
isoclinism	isoclinic class of #6631116
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1636 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26)(27,33)(28,34)(29,31)(30,38)(32,35)(36,40)(37,39);(3,7)(5,8)(10,14)(15,20)(16,21)(22,24)(26,31)(32,39)(33,40)(34,38);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,27,29,36)(26,32,31,39)(28,35,30,37)(33,34,40,38);(2,6)(3,7)(11,17)(15,20)(18,23)(22,24)(25,27)(26,33)(28,37)(29,36)(30,35)(31,40)(32,38)(34,39);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,28,29,30)(26,34,31,38)(27,35,36,37)(32,40,39,33)

Polycyclic presentation:

2^9|c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538612);