Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#21737 |
| 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 | C23×C5 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
662 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)(8,13)(9,11)(10,12)(14,15,21,25,19,24,20,36)(16,26,31,41,29,22,30,39)(17,32,35,40,34,44,18,23)(27,42,28,37,33,45,43,38);(6,8,10,9)(7,11,12,13)(14,16,19,29)(15,22,24,26)(17,33,34,27)(18,28,35,43)(20,30,21,31)(23,38,40,37)(25,39,36,41)(32,42,44,45);(14,17,20,18,19,34,21,35)(15,23,25,32,24,40,36,44)(16,27,30,43,29,33,31,28)(22,37,39,45,26,38,41,42);(6,9,10,8)(7,11,12,13)(14,18,20,34,19,35,21,17)(15,23,36,44,24,40,25,32)(16,28,30,27,29,43,31,33)(22,38,41,45,26,37,39,42)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14725);