Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#9875 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C2×D4 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle, prime radical | C23×C5 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C43×C5 ◁ G | |
434 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)(8,12)(9,10)(11,13)(14,15,19,20,18,21,17,16)(22,23)(24,30)(25,34)(26,28)(27,32)(29,31)(33,36)(35,37);(7,10)(11,12)(14,16,17,21,18,20,19,15)(22,24,31,37,26,33,25,32)(23,27,34,36,28,35,29,30);(6,8,9,13)(7,11,10,12)(14,17,18,19)(15,20,21,16)(23,28)(27,35)(29,34)(30,36)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2863);