#6580969

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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

1340 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)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,25)(19,27,31,23)(20,32,35,24)(22,26,36,43)(28,39,29,38)(30,40,34,42)(33,44,37,41)(45,47,46,48);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(19,28)(20,33)(22,36)(23,38)(24,41)(26,43)(27,39)(29,31)(32,44)(35,37)(45,46)(47,48);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,19)(18,23)(20,34)(21,31)(22,29)(24,42)(25,27)(26,39)(28,36)(30,35)(32,40)(33,46)(37,45)(38,43)(41,48)(44,47);(1,4,5,3)(2,8,6,7)(19,29)(20,33)(22,36)(23,39)(24,41)(26,43)(27,38)(28,31)(30,34)(32,44)(35,37)(40,42);(17,20,21,35)(18,24,25,32)(19,30,31,34)(22,37,36,33)(23,40,27,42)(26,44,43,41)(28,45,29,46)(38,47,39,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6488165);