#6516936

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500226
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3277 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=hi,g^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei

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, 6424132);