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#486681

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

Aliases

No aliases for this group.

Subgroups

678 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=gi,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dgi,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393877);