Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#13289 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#102 |
| 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 | |
342 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)(14,15,19,26,21,27,20,25)(16,28,33,45,34,44,32,40)(17,35,29,23,36,24,18,37)(22,39,41,31,43,30,42,38);(7,10)(11,12)(14,16,29,31,20,33,17,30,21,34,18,38,19,32,36,39)(15,22,24,44,26,42,37,45,27,43,35,28,25,41,23,40);(6,8,9,13)(7,11,10,12)(14,17,19,29,21,36,20,18)(15,23,25,35,27,37,26,24)(16,30,32,31,34,39,33,38)(22,40,41,28,43,45,42,44)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6277);