Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#22062 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C5×#3002 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C5×M16 ◁ G | |
900 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(14,15,18,22,19,17,26,21)(16,20,25,28,24,23,29,27);(6,8)(7,10)(9,13)(11,12)(14,16,18,25,19,24,26,29)(15,20,22,28,17,23,21,27);(8,13)(10,12)(14,17)(15,19)(16,23)(18,21)(20,24)(22,26)(25,27)(28,29);(7,11)(10,12)(16,24)(20,23)(25,29)(27,28)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15050);