#6586343

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1396 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gi,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6493539);