Vanguard Group is launching a private-equity fund, a striking move for the money-management giant that built a household brand on the back of low-cost funds for everyday investors.

The new fund is part of Chief Executive Tim Buckley’s push to broaden the Malvern, Pa.-based company’s appeal as a financial adviser for larger investors. With Mr. Buckley at the helm, the firm with $6.2 trillion in assets under management has pressed beyond the index-fund business it is best known for and further into the advisory business. It…

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