#6351828

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

Polycyclic presentation:

2^9|b^2=i,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, 6259024);