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#6817481

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

Aliases

No aliases for this group.

Subgroups

1542 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 5 generators on 80 points:

(1,2,6,10)(3,11,13,7)(4,8,5,9)(12,16,14,15)(17,18,23,29)(19,24,34,47)(20,35,37,52)(21,26,43,56)(22,28,41,27)(25,48,50,39)(30,49,58,73)(31,45,62,71)(32,57,33,46)(36,63,60,44)(38,64,66,77)(40,53,61,78)(42,55,69,54)(51,72,75,68)(59,74,67,80)(65,76,79,70);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,16)(10,11)(17,19)(18,24)(20,36)(21,40)(22,32)(23,34)(25,49)(26,53)(27,46)(28,57)(29,47)(30,39)(31,42)(33,41)(35,63)(37,60)(38,67)(43,61)(44,52)(45,55)(48,58)(50,73)(51,76)(54,71)(56,78)(59,66)(62,69)(64,80)(65,68)(70,75)(72,79)(74,77);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20,23,37)(18,25)(19,30)(21,38,43,66)(22,39,41,48)(24,44,47,63)(26,51)(27,35)(28,52)(29,50)(31,59)(32,60)(33,36)(34,58)(40,65)(42,68,69,72)(45,70,71,76)(46,49,57,73)(53,74,78,80)(54,64)(55,77)(56,75)(61,79)(62,67);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,16,11,15)(20,37)(25,50)(30,58)(35,52)(36,60)(38,66)(39,48)(44,63)(49,73)(51,75)(59,67)(64,77)(65,79)(68,72)(70,76)(74,80);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(17,21,41,69,23,43,22,42)(18,26,27,54,29,56,28,55)(19,31,32,61,34,62,33,40)(20,38,48,72,37,66,39,68)(24,45,57,78,47,71,46,53)(25,51,35,64,50,75,52,77)(30,59,60,79,58,67,36,65)(44,70,73,80,63,76,49,74)

Polycyclic presentation:

2^9|a^2=fi,c^2=fgh,d^2=f,e^2=fhi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6724677);