Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#11963 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#102 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| prime radical | C5×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7,8,11,9,10,12,13)(14,15,19,24,17,26,33,23,20,25,36,41,34,22,18,35)(16,27,30,39,32,40,45,43,31,42,44,21,28,38,29,37);(7,10)(8,12)(14,16)(15,21)(17,28)(18,30)(19,29)(20,31)(22,37)(23,40)(24,42)(25,39)(26,43)(27,41)(32,34)(33,44)(35,38)(36,45);(7,10)(11,13)(14,17,20,34)(15,22,25,26)(16,28,31,32)(18,19,33,36)(21,37,39,43)(23,24,35,41)(27,40,42,38)(29,44,45,30)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4951);