Blackstone Group LP is cashing in on its bet on the suburban rental class. : The private-equity firm late Tuesday sold more than $1 billion of shares of Invitation Homes Inc. , the giant single-family home landlord it launched following the financial crisis in a wager that many Americans would be willing to rent the suburban lifestyle they could no longer afford to own. The share offering is Blackstone’s second since March and comes as Invitation’s shares are trading at a record, reflecting rising rents and strong demand for the 80,000-odd homes it owns in 17 markets around the country.