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#6585064

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

Aliases

No aliases for this group.

Subgroups

1720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,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, 6492260);