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The video of Mr Perkins shows him and a group of people talking before gunshots are heard. The phone then appears to tumble through bloody grass before going black.
Bystanders can be heard screaming and crying.
The video remains on Facebook with a warning message about its graphic nature.
A spokeswoman for Facebook said the video does not violate company policy. The social media site would remove a video if it celebrated or glorified violence, she said.
Earlier this month, the killer of a French police commander and the commander's partner broadcast on Facebook Live shortly after the killings, urging followers to kill prison staff, police officials, journalists and lawmakers.
Chicago has one of the worst gun crime rates of any US city. There were nearly 500 homicides last year, and gun violence is up in 2016, police say.