Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#8601 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #117 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #380 |
| prime radical | C24×C5 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#97 ◁ G | |
462 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)(6,7,8,10,9,11,12,13)(14,15)(16,24)(17,28,30,21)(18,31)(19,23)(20,26)(22,32)(25,38,40,36)(27,37)(29,35,44,43)(33,39,42,45)(34,41);(14,16,18,26,19,27,32,41)(15,20,22,24,23,34,31,37)(17,25,29,39,30,40,44,45)(21,33,35,36,28,42,43,38);(14,17)(15,21)(16,25)(18,29)(19,30)(20,33)(22,35)(23,28)(24,36)(26,39)(27,40)(31,43)(32,44)(34,42)(37,38)(41,45)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1589);