5570 doesn't support eyefinity. the lowest card that does support that are some of the 5750 and 5770.
A review from Techspot.com
said, "We have to give it to AMD, however, it's a great idea to keep all features intact across the entire HD 5000 series. The Radeon HD 5570 can take advantage of triple monitor output, if not for gaming, Windows and productivity apps will work just fine."
page also states it supports 3 monitor setup. However, a single card can only support 2 monitors at a time. As pixar1942 said, you will need to crossfire the card to get the third display.
If I get around to it, I will test it on my AMD 890GX based board to see if it works (at least until I can buy a HD5770 or better).