In reply #7 of the Best method for getting to the next age? thread, Gilmoy laid down the challenge for the fastest first successful invasion.
What is the lower bound on the turn number of your 1st successful planetary invasion? A submission consists of two screenshots, showing the turn number, the planet while enemy-held (with your transport within 1 turns' range), and the planet after capture, in the same turn (which means you move your transport manually). The turn number automatically includes your transport's transit time, so ship speed and proximity to enemy helps.
Post your screenshots as requested by Gilmoy here.