#6579622

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×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1430 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,24,31,40)(20,25,32,41)(21,26,36,45)(22,27,30,39)(29,38,34,43)(33,42,47,48)(35,44,37,46);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(19,21)(20,32)(22,30)(24,26)(25,41)(27,39)(29,47)(31,36)(33,34)(38,48)(40,45)(42,43);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,19)(18,24)(20,33)(21,30)(22,36)(23,31)(25,42)(26,39)(27,45)(28,40)(29,37)(32,47)(34,35)(38,46)(41,48)(43,44);(17,20,23,32)(18,25,28,41)(19,29,31,34)(21,33,36,47)(22,37,30,35)(24,38,40,43)(26,42,45,48)(27,46,39,44);(17,21,23,36)(18,26,28,45)(19,30,31,22)(20,34,32,29)(24,39,40,27)(25,43,41,38)(33,35,47,37)(42,44,48,46)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=f,d^2=i,e^2=i,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, 6486818);