#6579617

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1405 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)(19,29)(20,25)(21,26)(22,27)(23,28)(24,32)(30,47)(31,45)(33,41)(34,44)(35,40)(36,43)(37,39)(38,46)(42,48);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(19,21)(20,33)(23,31)(24,39)(25,41)(26,29)(28,45)(30,48)(32,37)(34,35)(40,44)(42,47);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,19)(18,24)(20,34)(21,31)(22,32)(23,37)(25,42)(26,28)(27,29)(30,38)(33,48)(35,36)(39,45)(40,46)(41,44)(43,47);(17,20,22,33)(18,25,27,41)(19,30,32,35)(21,34,37,48)(23,38,31,36)(24,40,29,47)(26,44,39,42)(28,46,45,43);(17,21,22,37)(18,26,27,39)(19,31,32,23)(20,35,33,30)(24,28,29,45)(25,40,41,47)(34,36,48,38)(42,46,44,43)

Polycyclic presentation:

2^9|b^2=fgi,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, 6486813);