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#7501823

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

Aliases

No aliases for this group.

Subgroups

2276 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)(9,10,16,21)(11,17,25,42)(12,26,31,45)(13,32,35,47)(14,36,15,20)(18,29,46,30)(19,33,48,34)(22,49,52,43)(23,53,56,64)(24,41,37,40)(27,38,51,63)(28,44,60,68)(39,54,65,55)(50,62,71,72)(57,61,67,59)(58,66,70,69);(1,3,5,2)(4,7,8,6)(9,11)(10,17)(12,27)(13,23)(14,24)(15,37)(16,25)(18,43)(19,39)(20,40)(21,42)(22,29)(26,38)(28,58)(30,52)(31,51)(32,53)(33,54)(34,55)(35,56)(36,41)(44,66)(45,63)(46,49)(47,64)(48,65)(50,59)(57,62)(60,70)(61,71)(67,72)(68,69);(1,4,5,8)(2,6,3,7)(9,12)(10,18)(11,22)(13,28)(14,29)(15,30)(16,31)(17,38)(19,44)(20,45)(21,46)(23,50)(24,51)(25,52)(26,36)(27,37)(32,57)(33,59)(34,61)(35,60)(39,62)(40,43)(41,49)(42,63)(47,67)(48,68)(53,69)(54,70)(55,58)(56,71)(64,66)(65,72);(2,3)(6,7)(9,13)(10,19)(11,23)(12,28)(14,33)(15,34)(16,35)(17,39)(18,44)(20,47)(21,48)(22,50)(24,54)(25,56)(26,57)(27,58)(29,59)(30,61)(31,60)(32,36)(37,55)(38,62)(40,64)(41,53)(42,65)(43,66)(45,67)(46,68)(49,69)(51,70)(52,71)(63,72);(9,14,16,15)(10,20,21,36)(11,24,25,37)(12,29,31,30)(13,33,35,34)(17,40,42,41)(18,45,46,26)(19,47,48,32)(22,51,52,27)(23,54,56,55)(28,59,60,61)(38,43,63,49)(39,64,65,53)(44,67,68,57)(50,70,71,58)(62,66,72,69)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,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, 7409019);