Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#11001 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #379 |
| prime radical | C5×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(10,11)(12,21)(13,25,27,34)(14,28)(15,19)(16,20)(17,23)(18,26,35,30)(22,37,40,31)(24,32)(29,33)(36,39,41,38);(6,7)(8,9)(10,12)(11,17)(13,22)(14,29)(15,23)(16,24)(18,31)(19,32)(20,33)(21,28)(25,36)(26,39)(27,40)(30,38)(34,41)(35,37);(7,9)(10,13)(11,18)(12,22)(14,26)(15,30)(16,27)(17,31)(19,34)(20,35)(21,36)(23,38)(24,40)(25,28)(29,39)(32,41)(33,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3989);