Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#20829 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C22×C4⋊C4 |
| prime radical | C24×C5 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#117 ◁ G | |
718 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)(8,11)(9,10)(12,13)(14,15,19,23,20,24,18,34)(16,21,28,37,26,36,27,43)(17,29,32,39,33,45,31,22)(25,40,42,38,30,44,41,35);(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);(14,17,18,31,20,33,19,32)(15,22,23,29,24,39,34,45)(16,25,27,41,26,30,28,42)(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)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13817);