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#513137

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

Aliases

No aliases for this group.

Subgroups

295 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 4 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=e,d^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,h^a=hi,c^b=cei,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420333);