Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#21080 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#121 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#388 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,11,15,12,16,10,26)(8,17,21,28,19,31,20,13)(9,22,25,37,24,14,23,29)(18,32,35,27,33,30,34,36);(6,8)(7,13,16,28)(9,18)(10,20)(11,21)(12,19)(14,27,22,36)(15,17,26,31)(23,34)(24,33)(25,35)(29,30,37,32);(6,9,10,23,12,24,11,25)(7,14,15,29,16,22,26,37)(8,18,20,34,19,33,21,35)(13,27,17,30,28,36,31,32);(8,19)(13,28)(17,31)(18,33)(20,21)(27,36)(30,32)(34,35)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14068);