Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#26080 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7,9,11)(8,12,13,10)(14,15,19,25)(16,26,29,22)(17,30,32,44)(18,34,20,24)(21,35,37,28)(23,33,39,31)(27,41,43,36)(38,42,40,45);(14,16,18,28,19,29,20,35)(15,21,24,26,25,37,34,22)(17,27,31,42,32,43,33,45)(23,36,30,40,39,41,44,38);(6,8,9,13)(7,10,11,12)(14,16,19,29)(15,22,25,26)(17,27,32,43)(18,28,20,35)(21,34,37,24)(23,38,39,40)(30,36,44,41)(31,42,33,45);(14,17,18,31,19,32,20,33)(15,23,24,30,25,39,34,44)(16,27,28,42,29,43,35,45)(21,36,26,40,37,41,22,38)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19068);