#6807041

order	512
nilpotency class	3
isoclinism	isoclinic class of #6807031
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

1968 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)(3,7)(4,6)(5,8)(9,10)(11,22)(12,27)(13,30)(14,20)(15,21)(16,19)(17,26)(18,29)(23,39)(24,34)(25,38)(28,37)(31,36)(32,40)(33,35);(1,3)(2,5)(4,8)(6,7)(9,11,15,26)(10,17,21,22)(12,14,28,32)(13,24,16,33)(18,36,38,39)(19,34,30,35)(20,27,40,37)(23,25,31,29);(9,12,16,29,15,28,13,25)(10,18,19,37,21,38,30,27)(11,23,24,14,26,31,33,32)(17,20,35,36,22,40,34,39);(2,6)(5,7)(9,13,15,16)(10,19,21,30)(11,24,26,33)(12,25,28,29)(14,31,32,23)(17,34,22,35)(18,37,38,27)(20,39,40,36);(3,8)(5,7)(9,14,16,31,15,32,13,23)(10,20,19,36,21,40,30,39)(11,25,24,12,26,29,33,28)(17,18,35,37,22,38,34,27)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=i,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,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, 6714237);