#6359118

order	512
nilpotency class	3
isoclinism	isoclinic class of #6359112
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

2028 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)(11,21,25,38)(12,17,28,36)(13,19,14,18)(16,30,33,24)(22,32,26,40)(23,31,39,34)(27,37,29,35);(9,11,15,25)(10,16,20,33)(12,26,28,22)(13,24,14,30)(17,34,36,31)(18,21,19,38)(23,29,39,27)(32,37,40,35);(2,5)(3,8)(6,7)(9,12)(10,17)(11,22)(13,29)(14,27)(15,28)(16,31)(18,37)(19,35)(20,36)(21,32)(23,24)(25,26)(30,39)(33,34)(38,40);(9,12)(10,17)(11,23)(13,27)(14,29)(15,28)(16,32)(18,35)(19,37)(20,36)(21,31)(22,24)(25,39)(26,30)(33,40)(34,38);(9,13,15,14)(10,18,20,19)(11,24,25,30)(12,27,28,29)(16,21,33,38)(17,35,36,37)(22,39,26,23)(31,40,34,32)

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,e^c=eh,g^c=gi,f^d=fh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6266314);