#6359231

order	512
nilpotency class	3
isoclinism	isoclinic class of #6359226
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

1953 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6266427);