#6352950

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352941
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	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3257 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=hi,e^2=hi,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260146);