#6808116

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

1767 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=f,e^2=hi,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, 6715312);