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#6585046

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

Polycyclic presentation:

2^9|c^2=g,d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,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, 6492242);