#6590168

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1874 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6497364);