Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C24 |
| direct product | C5×#33494 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C24 |
| commutator | C22 | C23×C5×C8 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C22×C5 | C23×C8 |
| prime radical | C5×#532 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ G | |
556 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)(8,12)(9,11)(10,13)(14,15,17,22,18,23,28,36,19,24,29,37,30,38,41,45)(16,20,26,31,27,32,40,42,25,21,33,34,39,35,43,44);(6,8)(7,10)(9,13)(11,12)(14,16)(15,20)(17,26)(18,27)(19,25)(21,24)(22,31)(23,32)(28,40)(29,33)(30,39)(34,37)(35,38)(36,42)(41,43)(44,45);(8,13)(10,12)(14,16)(15,21)(17,26)(18,27)(19,25)(20,24)(22,34)(23,35)(28,40)(29,33)(30,39)(31,37)(32,38)(36,44)(41,43)(42,45);(7,11)(10,12)(16,25)(20,21)(26,33)(27,39)(31,34)(32,35)(40,43)(42,44)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26482);