Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33749 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C23×C5 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,19,23)(16,24,25,38)(17,29,31,44)(18,32,30,22)(20,33,34,27)(21,36,37,26)(28,43,41,40)(35,42,39,45);(7,10)(11,13)(16,25)(20,34)(24,38)(27,33)(28,41)(35,39)(40,43)(42,45);(14,16,26,42,18,28,31,33,19,25,36,45,30,41,17,27)(15,20,29,40,22,35,37,38,23,34,44,43,32,39,21,24);(6,8,9,12)(7,11,10,13)(16,25)(20,34)(24,38)(27,33)(28,41)(35,39)(40,43)(42,45)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26737);