There are specific missions on the bulletin board to rank up with Empire/Federation (and permits). Alliance ones give permits.
That's true - there are missions that specifically say you receive rep, but all general dealings of any kind with a station will contribute to rep with its controlling faction by default. For example, if you return from deep space exploration with a heap of data and sell it at a station where you have no influence after a certain amount of sales the controlling minor faction will turn friendly. The same goes for trading - after a certain amount of trading between stations each one will turn friendly towards you.
The missions allow you to pick up local rep by working for an opposing local faction as well as the controlling faction - eg Say there are 4 minor factions present at a station - two imperial, one independent and one federation, and one of the imperial factions is in control. All your general dealings will raise rep with the controlling faction, and so will their specific missions, particularly the rep missions. But you can also take missions for the other imperial minor faction and gain rep with them and the Empire. In some cases this may affect your standing with the controlling faction, but generally not too much.
The critical thing if you are working on rep is to make sure you only ever work for stations associated with the major faction you want to influence - even at the expense of making less than the best money on trades. Also never take missions for any factions that are opposed to the major faction - it will reduce your rep.