#6572550

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533063
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

2872 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

(17,18,22,26,23,20,21,25)(19,27,29,28,31,32,30,24);(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,19,23,31)(18,24,20,28)(21,29,22,30)(25,27,26,32);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,20)(18,23)(19,28)(21,26)(22,25)(24,31)(27,30)(29,32);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,21,23,22)(18,25,20,26)(19,29,31,30)(24,27,28,32);(4,12)(7,14)(10,16)(13,15)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6479746);