#6686977

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682628
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

1827 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=f,e^2=fi,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,h^c=hi,h^e=hi

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, 6594173);