#6590182

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1744 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)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15)(17,18,22,26)(19,23,29,39)(20,24,33,43)(21,25,36,46)(27,37,47,48)(28,38,34,44)(30,40,31,41)(32,42,35,45);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,22,29)(18,23,26,39)(20,30,33,31)(21,34,36,28)(24,40,43,41)(25,44,46,38)(27,35,47,32)(37,45,48,42);(1,3,5,4)(2,7,6,8)(10,15)(14,16)(17,20,22,33)(18,24,26,43)(19,27,29,47)(21,35,36,32)(23,37,39,48)(25,45,46,42)(28,31,34,30)(38,41,44,40);(9,12,13,11)(10,16,15,14)(17,19,22,29)(18,23,26,39)(20,31,33,30)(21,34,36,28)(24,41,43,40)(25,44,46,38)(27,32,47,35)(37,42,48,45);(1,4,5,3)(2,8,6,7)(19,28)(21,36)(23,38)(25,46)(27,31)(29,34)(30,47)(32,35)(37,41)(39,44)(40,48)(42,45)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fi,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=ehi,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6497378);