#6357346

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357317
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

1901 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,24,21)(19,26,31,32)(20,25,30,29)(22,27,23,28);(3,11)(7,15)(10,16)(12,13)(17,19,22,20,24,31,23,30)(18,25,28,26,21,29,27,32);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16)(17,20,22,31,24,30,23,19)(18,26,28,29,21,32,27,25);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(17,21,24,18)(19,29,31,25)(20,32,30,26)(22,27,23,28);(17,22,24,23)(18,27,21,28)(19,20,31,30)(25,32,29,26)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gh,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ci,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, 6264542);