Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#8248 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | C4×C22⋊C4 |
| prime radical | C5×#121 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C4×C5×C8 ◁ 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)(8,14)(9,17)(10,13)(11,16)(12,18)(15,20)(19,21)(22,23,27,30,28,24,26,32)(25,29,34,36,35,31,33,37);(6,8)(7,11,13,19)(9,16)(10,15)(12,14,17,20)(18,21)(22,24,28,23)(25,29,35,31)(26,30,27,32)(33,37,34,36);(8,15)(11,19)(14,20)(16,21)(22,25,27,34,28,35,26,33)(23,29,30,36,24,31,32,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1236);