#6580809

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
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#126 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1413 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)(11,13)(12,14)(15,16)(17,18,22,25)(19,23,29,20)(21,24,32,31)(26,27,30,28);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,18,22,25)(19,26,29,30)(20,28,23,27)(21,31,32,24);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,19)(18,23)(20,25)(21,27)(22,29)(24,30)(26,31)(28,32);(1,4,5,3)(2,8,6,7)(19,27)(20,26)(21,32)(23,30)(24,31)(28,29);(17,20,22,23)(18,19,25,29)(21,26,32,30)(24,27,31,28)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=ehi,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, 6488005);