The best way to put it is “phylum, order, genus”. The phylum is a larger classification. It groups organisms by a specific kind of characteristic. The order further narrows it down by characteristics. Genus is part of the bionomenclature that is used to give animals and plants their scientific names, so it defines the organism even further by a type of organism (such as “dog” or “cat”).
The diagram given is also a good way to show which order it should go in. While “genus, order, phylum” could be correct reading up the diagram, “phylum, order, genus” is also correct when reading down the diagram, and therefore reading down the page. However, both orders would be correct since it shows the relationship both ways.