#6573418

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533141
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3437 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 24 points:

(9,10)(11,14)(12,17,20,15)(13,18)(16,21)(19,22,24,23);(1,2,6,5)(3,8,4,7)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23);(1,3,6,4)(2,7,5,8)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23);(1,4,6,3)(2,7,5,8)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23);(1,5,6,2)(3,8,4,7)(13,21)(16,18)(19,24)(22,23)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=h,d^2=h,e^2=hi,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,g^b=gi,f^c=fi,e^d=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480614);