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#6579439

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

Aliases

No aliases for this group.

Subgroups

1854 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)(9,10,17,23)(11,24,29,46)(12,30,13,20)(14,21,39,55)(15,40,16,22)(18,41,47,28)(19,35,50,32)(25,44,59,43)(26,56,33,48)(27,57,62,52)(31,63,34,51)(36,45,61,70)(37,49,65,71)(38,54,42,53)(58,68,64,67)(60,72,66,69);(2,5)(6,8)(9,11,16,41,17,29,15,28)(10,18,40,46,23,47,22,24)(12,26,35,59,13,33,32,25)(14,36,42,27,39,61,38,62)(19,48,20,43,50,56,30,44)(21,52,54,70,55,57,53,45)(31,64,65,66,34,58,37,60)(49,68,51,69,71,67,63,72);(1,3,4,7)(2,6,5,8)(9,12,15,32,17,13,16,35)(10,19,22,30,23,50,40,20)(11,25,28,33,29,59,41,26)(14,31,38,37,39,34,42,65)(18,43,24,48,47,44,46,56)(21,49,53,63,55,71,54,51)(27,58,36,66,62,64,61,60)(45,67,57,69,70,68,52,72);(9,13)(10,20)(11,26)(12,17)(14,37)(15,35)(16,32)(18,44)(19,22)(21,49)(23,30)(24,56)(25,28)(27,58)(29,33)(31,42)(34,38)(36,66)(39,65)(40,50)(41,59)(43,47)(45,69)(46,48)(51,54)(52,67)(53,63)(55,71)(57,68)(60,61)(62,64)(70,72);(9,14)(10,21)(11,27)(12,31)(13,34)(15,38)(16,42)(17,39)(18,45)(19,49)(20,51)(22,53)(23,55)(24,57)(25,58)(26,60)(28,36)(29,62)(30,63)(32,37)(33,66)(35,65)(40,54)(41,61)(43,67)(44,68)(46,52)(47,70)(48,69)(50,71)(56,72)(59,64)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=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, 6486635);