#6362763

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

1747 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)(9,10,15,20)(11,21,22,35)(12,25,28,17)(13,19,14,18)(16,29,31,24)(23,33,38,37)(26,32,39,40)(27,34,30,36);(9,11,15,22)(10,16,20,31)(12,26,28,39)(13,24,14,29)(17,33,25,37)(18,21,19,35)(23,30,38,27)(32,36,40,34);(2,5)(3,8)(6,7)(11,22)(16,31)(21,35)(23,38)(24,29)(26,39)(32,40)(33,37);(9,12,15,28)(10,17,20,25)(11,23,22,38)(13,27,14,30)(16,32,31,40)(18,34,19,36)(21,37,35,33)(24,39,29,26);(2,6)(5,7)(9,13,15,14)(10,18,20,19)(11,24,22,29)(12,27,28,30)(16,21,31,35)(17,34,25,36)(23,39,38,26)(32,37,40,33)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,e^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, 6269959);