#6696341

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1473 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6603537);