Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12019 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#12887 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#106 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
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)(16,26,30,41)(17,32,31,45)(18,34)(19,35)(20,25)(21,29,39,42)(22,33,40,43)(23,28)(24,36)(27,38)(37,44);(7,10)(11,13)(14,16,27,42,19,31,44,33,20,30,18,29,36,17,28,43)(15,21,37,41,24,22,38,45,25,39,23,26,35,40,34,32);(6,8,9,12)(7,11,10,13)(14,17,18,33,19,16,28,29,20,31,27,43,36,30,44,42)(15,22,23,32,24,39,34,41,25,40,37,45,35,21,38,26)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5875);