#6352997

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352941
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×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2883 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^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=ehi

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, 6260193);