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#7624215

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

Aliases

No aliases for this group.

Subgroups

1493 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^9|a^2=i,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531411);