Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12832 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #427 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1807 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×M16 ◁ C2×C5×#131 ◁ G | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 69 points:
(1,2,3,4,5)(6,7)(8,18,21,39)(9,23,25,14)(10,27)(11,30)(12,16)(13,20,33,45)(15,28)(17,31)(19,41)(22,48,47,40)(24,52,53,56)(26,36,55,51)(29,57)(32,42)(34,61,59,46)(35,54,50,68)(37,58)(38,44)(43,62)(49,66)(60,69)(63,65)(64,67);(6,8,10,20,11,22,28,46,12,21,29,45,31,47,58,61)(7,13,15,18,17,34,27,40,16,33,37,39,30,59,57,48)(9,19,24,42,26,43,54,65,25,44,53,66,55,67,68,69)(14,32,35,38,36,60,56,62,23,49,50,41,51,63,52,64);(6,9)(7,14)(8,19)(10,24)(11,26)(12,25)(13,32)(15,35)(16,23)(17,36)(18,38)(20,42)(21,44)(22,43)(27,56)(28,54)(29,53)(30,51)(31,55)(33,49)(34,60)(37,50)(39,41)(40,62)(45,66)(46,65)(47,67)(48,64)(52,57)(58,68)(59,63)(61,69)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5820);