#6351821

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351817
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2860 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,g^a=ghi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259017);