#6681472

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1449 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6588668);