#6681746

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1471 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)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,21)(19,24,30,28)(20,25,33,34)(22,26,38,35)(27,39,46,47)(29,40,36,43)(31,41,45,48)(32,42,37,44);(1,3,6,2)(4,8,5,7)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,31)(21,28)(22,36)(23,30)(25,41)(26,43)(27,32)(29,38)(33,45)(34,48)(35,40)(37,46)(39,42)(44,47);(10,14)(13,16)(17,20,23,33)(18,25,21,34)(19,27,30,46)(22,37,38,32)(24,39,28,47)(26,44,35,42)(29,45,36,31)(40,48,43,41);(1,4,6,5)(2,7,3,8)(17,21,23,18)(19,28,30,24)(20,25,33,34)(22,35,38,26)(27,39,46,47)(29,43,36,40)(31,41,45,48)(32,42,37,44);(1,5,6,4)(2,7,3,8)(19,29)(22,38)(24,40)(26,35)(27,45)(28,43)(30,36)(31,46)(32,37)(39,48)(41,47)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=i,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6588942);