#6814814

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814811
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1761 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)(9,10,16,21)(11,17,23,30)(12,24,26,38)(13,19,31,33)(14,32,15,20)(18,29,36,28)(22,27,37,35)(25,34,39,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,25)(13,20)(14,19)(15,33)(16,23)(18,35)(21,30)(22,29)(24,34)(26,39)(27,36)(28,37)(31,32)(38,40);(9,12,16,26)(10,18)(11,22)(13,27,31,35)(14,28,15,29)(17,34,30,40)(19,25)(20,24)(21,36)(23,37)(32,38)(33,39);(2,6)(5,7)(12,26)(18,36)(22,37)(24,38)(25,39)(27,35)(28,29)(34,40);(3,8)(5,7)(9,13,15,17,16,31,14,30)(10,19,20,23,21,33,32,11)(12,27,29,34,26,35,28,40)(18,25,24,37,36,39,38,22)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di

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, 6722010);