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#7624424

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

Aliases

No aliases for this group.

Subgroups

1151 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531620);