it looks like some 570 models might be too large, but I bet you could squeeze one in there.
The main problem will be cooling, the fans will be basically up against the floor of the case and it will probably choke a lot. If you were daring, you could drill a ton of holes in the bottom of that case and add some feet to keep it off the ground, so that airflow can get to the card.
The other problem is that power supply. 18 amps on the 12V rail, I hope you plan on upgrading it.