Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#60685 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| socle | C22×C5 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#117 ◁ G | |
1878 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)(17,29)(18,23)(19,34)(20,24)(22,32)(25,40)(26,36)(27,37)(28,43)(30,44)(31,39)(33,45)(35,42)(38,41);(7,10)(11,13)(14,16)(15,21)(17,30)(18,27)(19,28)(20,26)(22,38)(23,37)(24,36)(25,33)(29,44)(31,42)(32,41)(34,43)(35,39)(40,45);(14,17,19,32,20,33,18,31)(15,22,23,29,24,39,34,45)(16,25,28,42,26,30,27,41)(21,35,37,40,36,38,43,44);(6,8,9,12)(7,11,10,13)(16,26)(21,36)(25,30)(27,28)(35,38)(37,43)(40,44)(41,42);(14,18,20,19)(15,23,24,34)(16,27,26,28)(17,31,33,32)(21,37,36,43)(22,29,39,45)(25,41,30,42)(35,40,38,44)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53673);