Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13240 |
| 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 | |
430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(7,9)(10,11,12,13)(14,15)(16,27)(17,32)(18,35)(19,37)(20,38)(21,26)(22,29)(23,33)(24,28)(25,36)(30,40)(31,45)(34,42)(39,43)(41,44);(6,7,8,9)(11,13)(14,16,28,43,20,31,18,29,21,17,33,44,36,34,19,30)(15,22,37,45,25,39,23,27,26,40,35,42,38,41,24,32);(14,17,33,43,20,34,19,29,21,16,28,44,36,31,18,30)(15,22,35,42,25,39,24,32,26,40,37,45,38,41,23,27)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6228);