#6356880

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

1616 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)(3,11)(4,8)(5,9)(6,14)(7,10)(12,15)(13,16)(18,22)(19,25)(20,28)(24,30)(26,27)(29,32);(1,3,5,12,7,13,6,8)(2,4,14,16,10,15,9,11)(17,18,20,22,21,24,28,30)(19,23,26,29,27,31,25,32);(17,19,21,27)(18,23,24,31)(20,26,28,25)(22,29,30,32);(1,4,7,15)(2,8,10,12)(3,9,13,14)(5,16,6,11)(18,24)(22,30)(23,31)(29,32);(1,5,7,6)(2,9,10,14)(3,12,13,8)(4,11,15,16)

Polycyclic presentation:

2^9|b^2=fhi,c^2=i,d^2=hi,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, 6264076);