#6360930

order	512
nilpotency class	3
isoclinism	isoclinic class of #6360896
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1756 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^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=chi,d^b=df,f^b=fh,e^c=eh,g^c=gi,f^d=fh

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, 6268126);