#6516931

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500157
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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3295 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,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, 6424127);