Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1366 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C2×C5×#1366 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #333 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| prime radical | C23×C5 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2)(35,36,37,38,39);(3,4,6,11,19,12,30,28,8,13,21,29,7,10,20,26)(5,14,16,27,22,25,18,33,15,31,32,34,17,9,23,24);(3,5,8,15)(4,9)(6,18,21,23)(7,22,19,17)(10,14)(11,27)(12,31)(13,25)(16,20,32,30)(24,28)(26,33)(29,34)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4943);