Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#25879 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3499 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
1178 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)(8,12)(9,11)(10,13)(14,15,20,24)(16,21,27,38)(17,28,30,41)(18,32,19,23)(22,34,40,31)(25,39,42,45)(26,37,33,36)(29,44,43,35);(6,8,10,9)(7,11,13,12)(14,16)(15,21)(17,29)(18,26)(19,33)(20,27)(22,39)(23,36)(24,38)(25,31)(28,44)(30,43)(32,37)(34,42)(35,41)(40,45);(6,8,10,9)(7,12,13,11)(14,17,18,31,20,30,19,34)(15,22,23,28,24,40,32,41)(16,25,26,43,27,42,33,29)(21,35,36,39,38,44,37,45);(6,9,10,8)(7,12,13,11)(17,30)(22,40)(25,42)(28,41)(29,43)(31,34)(35,44)(39,45)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18867);