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#7503719

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

Aliases

No aliases for this group.

Subgroups

1878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,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, 7410915);