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×#26851 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
720 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)(8,13)(9,11)(10,12)(14,15)(16,28)(17,34)(18,38)(19,26)(20,25)(21,27)(22,31)(23,36)(24,29)(30,44)(32,41)(33,40)(35,42)(37,43)(39,45);(6,8,10,9)(7,11,12,13)(14,16,20,32,21,33,19,31)(15,22,26,40,27,41,25,28)(17,35,36,29,37,18,30,39)(23,42,34,45,44,38,43,24);(6,9,10,8)(7,11,12,13)(14,17,19,30,21,37,20,36)(15,23,25,43,27,44,26,34)(16,29,31,35,33,39,32,18)(22,24,28,38,41,45,40,42);(7,12)(11,13)(14,18,19,29,21,35,20,39)(15,24,25,38,27,45,26,42)(16,30,31,37,33,36,32,17)(22,23,28,43,41,44,40,34)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19839);