#6362755

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362750
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

1750 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)(4,6)(7,8)(9,10,14,17)(11,19,23,34)(12,16,26,32)(13,27,28,18)(15,21,30,22)(20,35,24,38)(25,39,40,33)(29,36,31,37);(9,11,14,23)(10,15,17,30)(12,24,26,20)(13,21,28,22)(16,31,32,29)(18,34,27,19)(25,37,40,36)(33,35,39,38);(2,5)(3,8)(6,7)(9,12,14,26)(10,16,17,32)(11,20,23,24)(13,25,28,40)(15,29,30,31)(18,33,27,39)(19,35,34,38)(21,36,22,37);(10,17)(11,21)(13,28)(15,19)(16,32)(20,36)(22,23)(24,37)(25,40)(29,35)(30,34)(31,38);(2,6)(5,7)(10,17)(15,30)(16,32)(18,27)(19,34)(29,31)(33,39)(35,38)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ehi,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6269951);