#6356820

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356733
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1673 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)(3,12,15,16)(4,9,8,14)(5,10,6,13)(17,18)(19,21)(20,27)(22,25)(23,29)(24,26)(28,32)(30,31);(1,3,5,14,7,15,6,9)(2,8,13,16,11,4,10,12)(17,19,26,25,22,24,21,18)(20,27,29,32,30,31,28,23);(17,20,22,30)(18,23,25,32)(19,27,24,31)(21,28,26,29);(1,4)(2,9)(3,13)(5,12)(6,16)(7,8)(10,15)(11,14)(17,21,22,26)(18,19,25,24)(20,28,30,29)(23,27,32,31);(1,5,7,6)(2,10,11,13)(3,14,15,9)(4,16,8,12)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264016);