Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#8785 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C2×D4 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle, prime radical | C23×C5 | C2×C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C43×C5 ◁ G | |
382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,16,18,17,19,20,21)(22,23)(24,32)(25,28)(26,36)(27,31)(29,34)(30,33)(35,37);(7,10)(11,13)(22,24,33,35,27,34,26,25)(23,28,36,29,31,37,30,32);(6,8,9,12)(7,11,10,13)(22,25,26,34,27,35,33,24)(23,29,30,28,31,32,36,37)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1773);