#6590738

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1545 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=fi,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,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, 6497934);