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#6585836

• p-group
• nilpotent
• supersolvable
• metabelian
• solvable
• monomial
• stem

Aliases

No aliases for this group.

Subgroups

1852 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 5 generators on 72 points:

(1,2,5,6)(3,7,4,8)(9,10)(11,24)(12,30)(13,20)(14,21)(15,22)(16,40)(17,23)(18,28)(19,32)(25,57)(26,43)(27,56)(29,46)(31,63)(33,44)(34,51)(35,50)(36,67)(37,49)(38,53)(39,55)(41,47)(42,54)(45,61)(48,59)(52,62)(58,72)(60,68)(64,70)(65,71)(66,69);(2,6)(7,8)(9,11)(10,18)(12,26)(13,33)(14,36)(15,28)(16,29)(17,41)(19,48)(20,43)(21,52)(22,24)(23,46)(25,32)(27,42)(30,44)(31,64)(34,58)(35,59)(37,60)(38,62)(39,61)(40,47)(45,54)(49,69)(50,57)(51,70)(53,67)(55,56)(63,72)(65,66)(68,71);(1,3,5,4)(2,7,6,8)(9,12,17,35,15,13,16,32)(10,19,40,20,22,50,23,30)(11,25,41,26,28,59,29,33)(14,31,42,65,38,34,39,37)(18,43,47,57,24,44,46,48)(21,49,55,51,53,71,54,63)(27,58,62,60,61,64,36,66)(45,68,52,72,56,69,67,70);(1,3,5,4)(2,7,6,8)(9,13,15,12)(10,20,22,30)(11,26,28,33)(14,37,38,65)(16,35,17,32)(18,44,24,43)(19,40,50,23)(21,49,53,71)(25,41,59,29)(27,58,61,64)(31,39,34,42)(36,66,62,60)(45,70,56,72)(46,57,47,48)(51,54,63,55)(52,68,67,69);(1,4,5,3)(2,8,6,7)(9,14,15,38)(10,21,22,53)(11,27,28,61)(12,31,13,34)(16,39,17,42)(18,45,24,56)(19,49,50,71)(20,51,30,63)(23,54,40,55)(25,58,59,64)(26,60,33,66)(29,36,41,62)(32,37,35,65)(43,68,44,69)(46,67,47,52)(48,70,57,72)

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6493032);