Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#26995 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
762 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)(8,12)(9,11)(10,13)(14,15)(16,22)(17,30)(18,33)(19,26)(20,25)(21,28)(23,34)(24,31)(27,42)(29,37)(32,36)(35,41)(38,43)(39,44)(40,45);(6,8,9,13)(7,10,11,12)(14,16,19,28,20,17,31,29)(15,21,24,22,25,37,26,30)(18,27,34,43,35,32,45,44)(23,36,33,38,40,42,41,39);(7,11)(10,12)(14,17)(15,22)(16,20)(18,27)(19,29)(21,26)(23,39)(24,37)(25,30)(28,31)(32,35)(33,36)(34,43)(38,40)(41,42)(44,45);(7,11)(10,12)(14,18,19,34,20,35,31,45)(15,23,24,33,25,40,26,41)(16,27,28,43,17,32,29,44)(21,36,22,38,37,42,30,39)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19983);