Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9279 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9292 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C23×C5 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7,10,13,11,8,9,12)(14,15)(16,24)(17,28,30,21)(18,31)(19,23)(20,26)(22,33)(25,38,41,45)(27,39)(29,36,32,37)(34,40,44,42)(35,43);(6,8,11,7)(9,12,10,13)(14,16,18,26,19,27,33,43)(15,20,22,24,23,35,31,39)(17,25,32,42,30,41,29,40)(21,34,37,38,28,44,36,45);(6,7,11,8)(9,13,10,12)(14,17)(15,21)(16,25)(18,32)(19,30)(20,34)(22,37)(23,28)(24,38)(26,42)(27,41)(29,33)(31,36)(35,44)(39,45)(40,43)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2280);