#6351183

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351101
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 ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2706 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)(4,5)(8,14)(9,13)(11,12)(15,16)(18,23)(20,21)(24,31)(25,30)(26,32)(27,29);(1,2,6,9)(3,8,10,15)(4,7,5,13)(11,14,12,16)(17,18)(19,24)(20,23)(21,30)(22,25)(26,28)(27,32)(29,31);(1,3)(2,8)(4,11)(5,12)(6,10)(7,14)(9,15)(13,16)(17,18)(19,26)(20,23)(21,30)(22,25)(24,28)(27,31)(29,32);(3,10)(8,15)(11,12)(14,16)(17,19)(18,24)(20,29)(21,27)(22,28)(23,31)(25,26)(30,32);(1,4,6,5)(2,7,9,13)(3,11,10,12)(8,14,15,16)(17,20,22,21)(18,23,25,30)(19,27,28,29)(24,32,26,31)

Polycyclic presentation:

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

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, 6258379);