Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#60586 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| prime radical | C5×#3002 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#117 ◁ G | |
2024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 45 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15)(16,21)(17,23)(18,24)(19,33)(20,25)(22,30)(26,35)(27,36)(28,43)(29,37)(31,38)(32,39)(34,40)(41,44)(42,45);(7,10)(11,13)(14,16,20,29)(15,21,25,37)(17,26,31,41)(18,27,19,28)(22,34,39,45)(23,35,38,44)(24,36,33,43)(30,40,32,42);(14,17)(15,22)(16,26)(18,30)(19,32)(20,31)(21,34)(23,33)(24,38)(25,39)(27,40)(28,42)(29,41)(35,43)(36,44)(37,45);(6,8,9,12)(7,11,10,13)(14,17)(15,23)(16,26)(18,32)(19,30)(20,31)(21,35)(22,33)(24,39)(25,38)(27,42)(28,40)(29,41)(34,43)(36,45)(37,44);(14,18,20,19)(15,24,25,33)(16,27,29,28)(17,30,31,32)(21,36,37,43)(22,38,39,23)(26,40,41,42)(34,44,45,35)305 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53574);