Hi there! I think you're getting a little confused between reputation and influence. The % influence figures that you see listed are how much control that faction has over the current system. Your reputation is one of: Hostile, Unfriendly, Neutral, Friendly or Allied. You'll see this listed for each of the big three factions, and for all of the minor factions in your current system.
To get a permit for a system then you will need to be allied with the minor faction that controls the system you want a permit for. I'm not sure who it is for Sol, but they should have a few missions in systems nearby that you can start to grind rep for. Once you are up to Allied then you'll be able to take a mission that grants you the permit that you need.
You can also get rep by trading at a station owned by that faction, turning in bounties etc. Related factions, eg another Federal one, will help your standing with all minor factions within the same major faction as well, although not as much.
I did this a couple of days ago for my Alioth permit, and it took me quite a few missions over two days to get there. Probably about 15 missions for Alioth Independents, and another 50 for other alliance factions in the area.