#6350366

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350061
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6257562);