#6357038

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356901
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

1561 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)(4,9)(5,10)(6,14)(7,8)(12,16)(13,15)(18,22)(19,25)(20,29)(24,31)(27,28)(30,32);(1,3,5,13,7,12,6,9)(2,4,14,16,8,15,10,11)(17,18,20,22,21,24,29,31)(19,26,27,32,28,23,25,30);(2,8)(3,12)(9,13)(10,14)(17,19)(18,23)(20,27)(21,28)(22,30)(24,26)(25,29)(31,32);(1,4,7,15)(2,9,8,13)(3,10,12,14)(5,16,6,11)(18,24)(22,31)(23,26)(30,32);(1,5,7,6)(2,10,8,14)(3,13,12,9)(4,11,15,16)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264234);