#6681739

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	#42664	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1488 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)(17,18,22,20)(19,27,31,24)(21,33,35,25)(23,26,37,34)(28,32,39,30)(29,38,45,41)(36,47,48,40)(42,44,43,46);(1,3,6,2)(4,8,5,7)(9,11,12,15)(10,13,14,16)(17,19,22,31)(18,24,20,27)(21,29,35,45)(23,30,37,32)(25,38,33,41)(26,39,34,28)(36,44,48,46)(40,43,47,42);(10,14)(13,16)(17,20,22,18)(19,28,31,39)(21,25,35,33)(23,26,37,34)(24,30,27,32)(29,43,45,42)(36,47,48,40)(38,46,41,44);(1,4,6,5)(2,7,3,8)(17,21,22,35)(18,25,20,33)(19,29,31,45)(23,36,37,48)(24,38,27,41)(26,40,34,47)(28,42,39,43)(30,44,32,46);(1,5,6,4)(2,7,3,8)(19,30)(23,37)(24,39)(26,34)(27,28)(29,44)(31,32)(36,48)(38,43)(40,47)(41,42)(45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=i,d^2=f,e^2=fi,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, 6588935);