#6351121

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

2964 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)(17,18)(19,30)(20,26)(21,28)(22,27)(23,29)(24,31)(25,32);(1,2)(3,8)(4,7)(5,13)(6,9)(10,15)(11,14)(12,16)(17,19)(18,24)(20,21)(22,31)(23,32)(25,27)(26,29)(28,30);(1,3,6,10)(2,8,9,15)(4,11,5,12)(7,14,13,16)(17,20,23,31)(18,25,29,30)(19,21,32,22)(24,27,26,28);(3,10)(8,15)(11,12)(14,16)(17,20)(18,26)(19,21)(22,32)(23,31)(24,29)(25,27)(28,30);(1,4,6,5)(2,7,9,13)(3,11,10,12)(8,14,15,16)(17,21,23,22)(18,27,29,28)(19,20,32,31)(24,25,26,30)

Polycyclic presentation:

2^9|c^2=h,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, 6258317);