#6580551

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

1548 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)(17,18,21,23)(19,27,30,24)(20,25,33,36)(22,26,34,37)(28,38,35,39)(29,43,45,40)(31,42,32,41)(44,47,46,48);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(18,23)(19,28)(20,31)(22,34)(24,38)(25,41)(27,39)(30,35)(32,33)(36,42)(40,43)(44,46);(10,14)(13,16)(17,19,21,30)(18,24,23,27)(20,29,33,45)(22,35,34,28)(25,40,36,43)(26,38,37,39)(31,44,32,46)(41,47,42,48);(1,4,5,3)(2,6,7,8)(19,28)(20,32)(22,34)(24,39)(25,41)(26,37)(27,38)(29,45)(30,35)(31,33)(36,42)(40,43);(17,20,21,33)(18,25,23,36)(19,29,30,45)(22,32,34,31)(24,40,27,43)(26,41,37,42)(28,44,35,46)(38,47,39,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=i,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=ehi,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, 6487747);