#6586121

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1484 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,15,16,13)(17,18,22,23)(19,24,27,21)(20,25,29,31)(26,28,32,30);(1,3,5,4)(2,6,7,8)(18,23)(19,24)(21,27)(25,31)(26,28)(30,32);(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,28)(21,23)(22,27)(25,32)(26,31)(29,30);(1,4,5,3)(2,6,7,8)(17,20,22,29)(18,25,23,31)(19,26,27,32)(21,30,24,28);(17,21,22,24)(18,19,23,27)(20,26,29,32)(25,28,31,30)

Polycyclic presentation:

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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6493317);