It's based on a real group, also the Sabini's from London were the original Italian gang from South London
Yea the Peaky Blinders were real, although they appear to have exaggerated their power and organisation somewhat, by all historical accounts the Blinders were a relatively disparate, if large and notably violent, street gang similar to the Manchester Scuttlers, also they were at their peak (no pun intended) around the end of the 19th century rather than the 1920s.
If you spend time in some of the old canalside and lock pubs in and around Brum and the Black Country (where I grew up) you will still occasionally come across some old git who's willing to ramble off some handed-down tales of the Peaky Blinders, 'though God knows how accurate they'll likely be!
Worst aspect of the show, particularly the first series, were those God-awful comedy Midlands accents, they were absolutely atrocious!