#6678017

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#35687	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2226 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)(17,18,21,25)(19,27,30,23)(20,31,33,24)(22,26,34,40)(28,41,35,37)(29,38,45,43)(32,42,36,39)(44,47,46,48);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(19,28)(20,32)(22,34)(23,37)(24,39)(26,40)(27,41)(30,35)(31,42)(33,36)(44,46)(47,48);(10,14)(13,16)(17,19,21,30)(18,23,25,27)(20,29,33,45)(22,35,34,28)(24,38,31,43)(26,41,40,37)(32,44,36,46)(39,47,42,48);(2,6)(5,7)(20,33)(24,31)(29,45)(32,36)(38,43)(39,42)(44,46)(47,48);(3,8)(5,7)(17,20,21,33)(18,24,25,31)(19,29,30,45)(22,36,34,32)(23,38,27,43)(26,42,40,39)(28,44,35,46)(37,47,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,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, 6585213);