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#6579421

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,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=eh,h^b=hi,d^c=dh,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, 6486617);