#6352640

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352581
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×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2842 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:

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

Polycyclic presentation:

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

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, 6259836);