#6358592

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358586
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1979 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)(3,5)(4,7)(6,8)(9,10,15,20)(11,21,22,35)(12,17,27,34)(13,19,14,18)(16,28,30,24)(23,37,38,32)(25,40,39,31)(26,36,29,33);(9,11,15,22)(10,16,20,30)(12,25,27,39)(13,24,14,28)(17,32,34,37)(18,21,19,35)(23,29,38,26)(31,36,40,33);(2,5)(3,8)(6,7)(11,22)(16,30)(21,35)(23,38)(24,28)(25,39)(31,40)(32,37);(9,12)(10,17)(11,23)(13,26)(14,29)(15,27)(16,31)(18,33)(19,36)(20,34)(21,37)(22,38)(24,39)(25,28)(30,40)(32,35);(9,13,15,14)(10,18,20,19)(11,24,22,28)(12,26,27,29)(16,21,30,35)(17,33,34,36)(23,39,38,25)(31,37,40,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265788);