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#6585831

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

Aliases

No aliases for this group.

Subgroups

1852 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fgi,d^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,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, 6493027);