#6596335

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

3087 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,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, 6503531);