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#513010

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

Aliases

No aliases for this group.

Subgroups

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

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=ei,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^c=di,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420206);