#6811135

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806815
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1785 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=fi,e^2=fhi,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,g^c=gi

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, 6718331);