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#7514907

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

Aliases

No aliases for this group.

Subgroups

1853 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 5 generators on 72 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,g^c=gi,h^c=hi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7422103);