#6491432

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470067
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 ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3006 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:

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

Polycyclic presentation:

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

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, 6398628);