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#513203

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

Aliases

No aliases for this group.

Subgroups

353 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420399);