#6595657

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×#453
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×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3152 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)(11,19)(12,20)(13,17)(14,24)(15,18)(16,22)(21,23);(1,2,4,5)(3,6,7,8)(9,11,15,16)(10,12,18,23)(13,20,14,21)(17,22,24,19);(2,5)(3,7)(9,12)(10,16)(11,18)(13,22)(14,19)(15,23)(17,21)(20,24);(1,3)(2,6)(4,7)(5,8)(9,13,15,14)(10,17,18,24)(11,20,16,21)(12,22,23,19);(2,5)(3,7)(9,13,15,14)(10,17,18,24)(11,20,16,21)(12,22,23,19)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6502853);