#6357713

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357668
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

1872 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264909);