#6357348

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,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, 6264544);