Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13322 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 73 points:
(1,2,3,4,5)(10,11,17,23)(12,24,29,55)(13,30,33,52)(14,34,39,50)(15,22,16,21)(18,41,45,26)(19,40,47,32)(20,37,51,38)(25,56,59,73)(27,48,28,54)(31,49,36,46)(35,43,61,44)(42,65,72,68)(53,62,70,58)(57,63,60,71)(64,69,67,66);(6,7)(8,9)(10,12,17,29)(11,18,23,45)(13,25,33,59)(14,35,39,61)(15,28,16,27)(19,42,47,72)(20,48,51,54)(21,44,22,43)(24,50,55,34)(26,37,41,38)(30,63,52,71)(31,65,36,68)(32,62,40,58)(46,73,49,56)(53,64,70,67)(57,69,60,66);(7,9)(10,13)(11,19)(12,25)(14,36)(15,40)(16,32)(17,33)(18,42)(20,49)(21,52)(22,30)(23,47)(24,53)(26,60)(27,62)(28,58)(29,59)(31,39)(34,67)(35,68)(37,66)(38,69)(41,57)(43,63)(44,71)(45,72)(46,51)(48,56)(50,64)(54,73)(55,70)(61,65)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6310);