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) = C24×C4 |
| direct product | C5×#22283 |
| 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×#3150 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#97 ◁ G | |
826 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)(8,11)(9,12)(10,13)(14,15,19,23,20,24,18,28)(16,21,26,29,25,22,17,27);(6,8,11,7)(9,12,10,13)(14,16,19,26,20,25,18,17)(15,21,23,29,24,22,28,27);(6,9,11,10)(7,12,8,13)(14,17,18,25,20,26,19,16)(15,22,23,27,24,21,28,29);(6,10,11,9)(7,12,8,13)(16,25)(17,26)(21,22)(27,29)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15271);