#6353131

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353061
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

3009 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)(17,18,21,23)(19,25,27,32)(20,28,30,24)(22,29,31,26);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(20,30)(24,28)(25,32)(26,29);(1,4)(2,7)(3,9)(5,11)(6,13)(8,14)(10,15)(12,16)(18,23)(22,31)(24,28)(25,32);(4,11)(7,14)(9,15)(13,16)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260327);