Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#11029 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #121 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C23×C4×C5 ◁ G | |
1242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7)(8,9)(14,15,19,25,20,23,18,24)(16,26,29,37,30,21,28,38)(17,31,34,40,35,22,33,41)(27,36,43,45,44,42,32,39);(7,9)(10,11)(12,13)(14,16,20,30)(15,21,23,26)(17,32,35,43)(18,29,19,28)(22,39,31,45)(24,38,25,37)(27,34,44,33)(36,41,42,40);(11,13)(14,17,20,35)(15,22,23,31)(16,27,30,44)(18,34,19,33)(21,36,26,42)(24,41,25,40)(28,43,29,32)(37,45,38,39)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4017);