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#7514923

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=i,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=cgi,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, 7422119);