#6581548

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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1544 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,24)(19,27,31,23)(20,25,33,38)(22,26,35,39)(28,36,30,42)(29,43,45,37)(32,41,34,40)(44,47,46,48);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19,21,31)(18,23,24,27)(20,29,33,45)(22,28,35,30)(25,37,38,43)(26,42,39,36)(32,44,34,46)(40,47,41,48);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(18,24)(19,28)(20,32)(22,35)(23,36)(25,40)(27,42)(30,31)(33,34)(37,43)(38,41)(44,46);(1,4,5,3)(2,6,8,7)(17,20,21,33)(18,25,24,38)(19,29,31,45)(22,32,35,34)(23,37,27,43)(26,41,39,40)(28,44,30,46)(36,47,42,48);(19,30)(20,32)(22,35)(23,36)(25,41)(26,39)(27,42)(28,31)(29,45)(33,34)(37,43)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fi,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6488744);