#6586337

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1337 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)(17,18,22,20)(19,27,30,24)(21,25,32,28)(23,26,29,31);(2,6)(7,8)(17,19,22,30)(18,24,20,27)(21,23,32,29)(25,31,28,26);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20,22,18)(19,28,30,25)(21,24,32,27)(23,26,29,31);(1,4,5,3)(2,8,6,7)(17,21,22,32)(18,25,20,28)(19,29,30,23)(24,26,27,31);(19,21)(23,29)(24,28)(25,27)(26,31)(30,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,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

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6493533);