Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#13281 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C2×D8 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5)(6,7,8,9)(10,11)(12,16)(13,14)(15,17)(18,19)(20,30)(21,27)(22,37)(23,39)(24,29)(25,32)(26,35)(28,38)(31,44)(33,40)(34,46)(36,43)(41,48)(42,47)(45,49);(7,9)(11,14)(15,16)(18,20,21,31,23,33,35,47,24,34,36,48,38,49,22,32)(19,25,37,45,28,41,43,46,29,42,26,40,39,44,27,30);(10,12,13,17)(11,15,14,16)(18,21,23,35,24,36,38,22)(19,26,28,27,29,37,39,43)(20,31,33,47,34,48,49,32)(25,40,41,30,42,45,44,46)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6269);