#6810887

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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1675 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=f,e^2=fh,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, 6718083);