I'm not sure if you can squeeze any more out of that budget, but here's a whole new build for $595:
PCPartPicker part list:
http://pcpartpicker.com/p/3rt06
Price breakdown by merchant: http://pcpartpicker.com/p/3rt06/by_merchant/
Benchmarks: http://pcpartpicker.com/p/3rt06/benchmarks/
CPU: AMD FX-6300 3.5GHz 6-Core Processor ($109.99 @ Newegg)
CPU Cooler: Cooler Master Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler ($35.98 @ Newegg)
Motherboard: Asus M5A97 R2.0 ATX AM3+ Motherboard ($89.79 @ Newegg)
Memory: G.Skill Ripjaws X Series 8GB (2 x 4GB) DDR3-1600 Memory ($74.99 @ Newegg)
Storage: Western Digital Caviar Blue 500GB 3.5" 7200RPM Internal Hard Drive ($49.58 @ Amazon)
Video Card: Asus GeForce GTX 650 Ti 2GB Video Card ($150.66 @ Newegg)
Case: NZXT Source 210 (Black) ATX Mid Tower Case ($37.99 @ Micro Center)
Power Supply: Corsair Builder 500W 80+ Bronze Certified ATX Power Supply ($44.99 @ Micro Center)
Optical Drive: LG GH24NSB0 DVD/CD Writer ($15.98 @ OutletPC)
Total: $594.95
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2014-04-14 04:55 EDT-0400)
If it absolutely has to be under $566, here's the same build with 4GB of RAM instead of 8GB:
http://pcpartpicker.com/p/3rt1w - $560