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×#22282 |
| 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×#3147 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#97 ◁ G | |
810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7)(8,11)(9,12)(10,13)(14,15,19,23,20,24,18,33)(16,21,28,36,26,35,27,42)(17,29,31,37,32,43,30,22)(25,38,41,44,39,45,40,34);(6,8,11,7)(9,12,10,13)(14,16,19,28,20,26,18,27)(15,21,23,36,24,35,33,42)(17,25,31,41,32,39,30,40)(22,34,29,38,37,44,43,45);(6,9,11,10)(7,12,8,13)(14,17,18,30,20,32,19,31)(15,22,23,29,24,37,33,43)(16,25,27,40,26,39,28,41)(21,34,36,38,35,44,42,45);(6,10,11,9)(7,12,8,13)(16,26)(21,35)(25,39)(27,28)(34,44)(36,42)(38,45)(40,41)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15270);