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#6817491

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

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,e^c=ei

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, 6724687);