Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#12651 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #442 |
| commutator | C8 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1762 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C5×#97 ◁ G | |
412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,17,13,18,12,28)(8,19,22,34,23,35,26,30)(9,15,11,27,25,33,24,16)(14,20,29,21,32,36,31,37);(6,8,11,20,10,26,25,37,13,23,24,36,12,22,9,21)(7,14,15,30,28,29,16,19,18,32,33,34,17,31,27,35);(6,9,10,11,13,25,12,24)(7,15,17,27,18,33,28,16)(8,20,22,21,23,36,26,37)(14,19,29,34,32,35,31,30)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5639);