#6580326

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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1574 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)(17,18,21,25)(19,23,29,40)(20,31,34,24)(22,26,36,45)(27,47,48,38)(28,39,30,41)(32,43,33,42)(35,46,37,44);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,21,29)(18,23,25,40)(20,32,34,33)(22,28,36,30)(24,42,31,43)(26,39,45,41)(27,37,48,35)(38,46,47,44);(10,14)(13,16)(17,20,21,34)(18,24,25,31)(19,27,29,48)(22,37,36,35)(23,38,40,47)(26,46,45,44)(28,32,30,33)(39,42,41,43);(1,4,5,3)(2,6,7,8)(17,19,21,29)(18,23,25,40)(20,33,34,32)(22,28,36,30)(24,43,31,42)(26,39,45,41)(27,35,48,37)(38,44,47,46);(19,28)(20,34)(22,36)(23,39)(24,31)(26,45)(27,33)(29,30)(32,48)(38,43)(40,41)(42,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=i,d^2=fi,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, 6487522);