Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#21140 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#97 ◁ G | |
764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,19,24,20,23,18,32)(16,21,28,35,26,36,27,42)(17,29,30,22,31,38,33,37)(25,39,41,34,40,44,43,45);(7,10)(11,13)(14,16,19,28,20,26,18,27)(15,21,24,35,23,36,32,42)(17,25,30,41,31,40,33,43)(22,34,38,44,37,45,29,39);(14,17)(15,22)(16,25)(18,33)(19,30)(20,31)(21,34)(23,37)(24,38)(26,40)(27,43)(28,41)(29,32)(35,44)(36,45)(39,42);(6,8,9,12)(7,11,10,13)(15,23)(16,26)(22,37)(24,32)(25,40)(27,28)(29,38)(41,43)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14128);