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#486672

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

Aliases

No aliases for this group.

Subgroups

692 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)(19,33)(20,39)(21,29)(22,46)(23,31)(24,30)(25,32)(26,36)(27,42)(28,44)(34,62)(35,52)(37,69)(38,57)(40,71)(41,74)(43,59)(45,58)(47,60)(48,56)(49,51)(50,63)(53,68)(54,73)(55,72)(61,65)(64,70)(66,79)(67,76)(75,78)(77,80);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19)(18,26)(20,34)(21,36)(22,47)(23,37)(24,49)(25,38)(27,50)(28,57)(29,51)(30,52)(31,60)(32,53)(33,46)(35,44)(39,70)(40,64)(41,75)(42,65)(43,67)(45,68)(48,66)(54,79)(55,76)(56,78)(58,69)(59,77)(61,71)(62,74)(63,72)(73,80);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,20)(18,27)(19,34)(21,40)(22,41)(23,48)(24,42)(25,43)(26,50)(28,54)(29,55)(30,59)(31,39)(32,56)(33,61)(35,63)(36,64)(37,66)(38,67)(44,72)(45,73)(46,71)(47,75)(49,65)(51,76)(52,77)(53,78)(57,79)(58,74)(60,70)(62,69)(68,80);(17,21,24,44,25,45,23,22)(18,28,31,29,32,46,30,58)(19,35,37,36,38,47,49,68)(20,40,42,72,43,73,48,41)(26,51,52,57,53,69,60,33)(27,54,39,55,56,71,59,74)(34,63,66,64,67,75,65,80)(50,76,77,79,78,62,70,61)

Polycyclic presentation:

2^9|a^2=h,d^2=g,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393868);