#6352636

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352581
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3812 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=hi,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259832);