Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24290 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
1294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,18)(13,26)(14,29)(15,22)(16,21)(17,24)(19,30)(20,27)(23,37)(25,33)(28,32)(31,36)(34,38)(35,41)(39,40);(10,12,15,24,16,13,27,25)(11,17,20,18,21,33,22,26)(14,23,30,38,31,28,41,39)(19,32,29,34,35,37,36,40);(7,9)(10,13)(11,18)(12,16)(14,28)(15,25)(17,22)(19,34)(20,33)(21,26)(23,31)(24,27)(29,37)(30,39)(32,36)(35,40)(38,41);(10,14,27,41,16,31,15,30)(11,19,22,36,21,35,20,29)(12,23,25,39,13,28,24,38)(17,32,26,40,33,37,18,34)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17278);