#6357233

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357223
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1900 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264429);