Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9781 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#9789 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #390 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C22×C5×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,14,11,15,9,12)(8,16,18,13,19,20,17,21)(22,23,26,29,25,28,32,34)(24,27,31,36,30,35,33,37);(7,12,15,14)(13,16,21,20)(22,24,26,33,25,30,32,31)(23,27,34,36,28,35,29,37);(6,8,9,17,11,19,10,18)(7,13,14,20,15,21,12,16)(23,28)(27,35)(29,34)(36,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2777);