#6580333

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1555 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)(17,18,22,27)(19,24,31,41)(20,33,37,25)(21,26,35,43)(23,28,36,45)(29,44,47,39)(30,40,34,48)(32,42,38,46);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,34)(21,29)(22,31)(23,38)(25,40)(26,44)(27,41)(28,46)(30,37)(32,36)(33,48)(35,47)(39,43)(42,45);(10,14)(13,16)(17,20,22,37)(18,25,27,33)(19,29,31,47)(21,23,35,36)(24,39,41,44)(26,45,43,28)(30,38,34,32)(40,42,48,46);(1,4,5,3)(2,6,7,8)(20,35)(21,37)(23,36)(25,26)(28,45)(29,30)(32,38)(33,43)(34,47)(39,48)(40,44)(42,46);(17,21,22,35)(18,26,27,43)(19,30,31,34)(20,36,37,23)(24,40,41,48)(25,28,33,45)(29,32,47,38)(39,46,44,42)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=i,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,e^c=ei,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, 6487529);