The Copyright Enforcement Group is a company that has pursued copyright infringers in the past on behalf of various copyright holders. How "legitimate" all of their copyright claims are, however, is something I, once again, cannot tell you. I know that they have sent settlement notices and have collected settlement amounts from infringers and they have filed suits against a couple of identified John Does in the Does' home states. Google "Dunlap, Grubb & Weaver" for more information.
If you choose to pay the offered settlement amount to end the matter now, I advise that you do so only after reviewing the notice you received with an attorney, and that you do not send off money without obtaining a release from the copyright holder that states that there is no further legal action pending against you on this copyright infringement claim nor will any further legal action be taken against you on this copyright infringement claim. The release should also be reviewed by an attorney.
It is possible that you may receive more than one notice of infringement if you downloaded the same copyrighted material several times or illegally downloaded additional copyrighted material.
There may be attorneys who will review all of this for little or no charge, but you must check around your area (perhaps a law school or a legal aid clinic can be of assistance?). I know in Michigan there are attorneys who volunteer their time to help those with their legal issues - sometimes these are scheduled for a specific day in a courtroom and sometimes these are radio or television call-in programs. I don't know if your state (which you failed to mention) offers similar free personal reviews.
Good luck.