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#488755

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

Aliases

No aliases for this group.

Subgroups

769 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=i,d^2=ghi,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395951);