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×#60581 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| prime radical | C5×#3005 | 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)(14,15,19,24)(16,20,27,34)(17,22,28,36)(18,21,30,23)(25,32,38,43)(26,31,40,33)(29,35,41,37)(39,42,45,44);(7,10)(11,13)(14,16,19,27)(15,20,24,34)(17,25,28,38)(18,26,30,40)(21,31,23,33)(22,32,36,43)(29,39,41,45)(35,42,37,44);(15,21)(18,30)(20,31)(22,35)(23,24)(26,40)(29,41)(32,42)(33,34)(36,37)(39,45)(43,44);(6,8,9,12)(7,11,10,13)(14,17)(15,22)(16,25)(18,29)(19,28)(20,32)(21,35)(23,37)(24,36)(26,39)(27,38)(30,41)(31,42)(33,44)(34,43)(40,45);(17,28)(22,36)(25,38)(29,41)(32,43)(35,37)(39,45)(42,44)305 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53569);