#6358847

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358842
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

1975 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)(9,10,14,17)(11,19,21,35)(12,16,25,33)(13,26,29,18)(15,27,31,22)(20,37,38,32)(23,40,39,30)(24,36,28,34);(9,11)(10,15)(12,23)(13,27)(14,21)(16,32)(17,31)(18,35)(19,26)(20,24)(22,29)(25,39)(28,38)(30,34)(33,37)(36,40);(2,5)(3,8)(6,7);(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|a^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,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, 6266043);