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) = C23 |
| direct product | C5×#8086 |
| 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 | C2×#95 |
| prime radical | C24×C5 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C42×C5 ◁ G | |
522 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)(8,15)(9,13)(10,19)(11,14)(12,16)(17,20)(18,21)(22,23,25,29,27,30,26,35)(24,31,33,37,32,28,34,36);(6,8)(7,12,14,21)(9,18)(10,16)(11,17)(13,15,19,20)(22,24,26,34,27,32,25,33)(23,28,29,36,30,31,35,37);(6,9,11,10)(7,13,14,19)(8,16,17,18)(12,20,21,15)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1074);