#6351533

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351503
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

3088 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258729);