Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#8157 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | C4×C22⋊C4 |
| prime radical | C24×C5 | C42 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(14,15,18,22,19,23,30,36)(16,24,27,38,26,20,31,33)(17,21,28,34,29,35,41,44)(25,37,40,45,39,32,42,43);(6,7,8,11,9,10,12,13)(14,16)(15,20,23,24)(17,25)(18,27)(19,26)(21,32,35,37)(22,33,36,38)(28,40)(29,39)(30,31)(34,43,44,45)(41,42);(7,10)(11,13)(14,17,18,28,19,29,30,41)(15,21,22,34,23,35,36,44)(16,25,27,40,26,39,31,42)(20,32,33,43,24,37,38,45)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1145);