#6579643

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1411 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)(19,24)(20,30)(21,23)(22,26)(25,33)(27,42)(28,47)(29,38)(31,44)(32,43)(34,46)(35,39)(36,40)(37,45)(41,48);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(18,23)(19,27)(22,35)(24,40)(25,30)(28,32)(29,36)(31,48)(34,37)(38,42)(41,43)(44,47);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,19,21,29)(18,24,23,38)(20,31,33,32)(22,36,35,27)(25,43,30,44)(26,40,39,42)(28,37,48,34)(41,46,47,45);(17,20,21,33)(18,25,23,30)(19,28,29,48)(22,37,35,34)(24,41,38,47)(26,46,39,45)(27,32,36,31)(40,43,42,44);(17,19,21,29)(18,24,23,38)(20,32,33,31)(22,36,35,27)(25,44,30,43)(26,40,39,42)(28,34,48,37)(41,45,47,46)

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,d^a=di,c^b=ch,e^b=eh,h^b=hi,d^c=dh,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, 6486839);