Any actions (trading, selling exploration data, doing missions, handing in bounty vouchers) to minor faction part of the major faction will increase your reputation with major faction. Note that rep over 75% will decay back to 75% (from allied to friendly) if you don't keep doing actions that increase rep, same from other end from hostile to unfriendly if you stop being bad boy.
Why you need rep with major faction? There is 0 benefits for that. If you are after military rank progression only missions count towards that.