Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#26848 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
718 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)(16,27)(17,32)(18,36)(19,38)(20,25)(21,26)(22,30)(23,34)(24,37)(28,44)(29,43)(31,40)(33,45)(35,42)(39,41);(6,8,10,9)(7,11,12,13)(14,16,19,30,21,31,20,39)(15,22,25,27,26,41,38,40)(17,33,34,28,35,18,29,37)(23,36,32,44,43,45,42,24);(6,9,10,8)(7,11,12,13)(14,17,19,34,21,35,20,29)(15,23,25,32,26,43,38,42)(16,28,30,18,31,37,39,33)(22,24,27,36,41,44,40,45);(7,12)(11,13)(14,18,19,37,21,33,20,28)(15,24,25,36,26,44,38,45)(16,29,30,17,31,34,39,35)(22,23,27,32,41,43,40,42)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19836);