Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#13285 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×D8 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×#102 |
| prime radical | C23×C5 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
350 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)(8,12)(9,10)(11,13)(14,15,20,25,21,26,19,36)(16,27,31,43,32,44,30,39)(17,24,18,35,34,38,33,23)(22,37,40,45,41,28,42,29);(7,10)(11,12)(14,16,18,29,20,31,34,37,21,32,33,45,19,30,17,28)(15,22,38,44,36,42,35,43,26,41,24,27,25,40,23,39);(6,8,9,13)(7,11,10,12)(14,17,19,33,21,34,20,18)(15,23,25,24,26,35,36,38)(16,28,30,45,32,37,31,29)(22,39,40,27,41,43,42,44)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6273);