#6356763

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

1572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=i,d^2=i,e^2=i,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, 6263959);