PARIS, Nov 3 (Reuters) - French public investment bank Bpifrance said on Tuesday it had bought 5.08% of chemicals group Arkema in the first investment of its new equity investment fund Lac1 that was set up to boost French firms.

Bpifrance has so far deployed 348 million euros ($405.7 million) from the 4.2 billion fund building up the stake since June and aims to keep adding to the position, Chief Executive Nicolas Dufourcq told journalists.

Bpifrance, which manages the fund, has put 1 billion euros of its own money into Lac1 with the aim of shoring up the capital base of big French companies with long-term investments.

Abu Dhabi sovereign wealth fund Mubadala Investment Co has also put 1 billion euros into the fund while French insurers and other investors have committed 1.2 billion euros. The fund has also raised 1 billion euros in debt for its investments.

Bpifrance said in the past it eventually aimed to raise up to 10 billion euros for the fund and invest in about 15 companies.

Dufourcq said Bpifrance was looking at other companies and was close to taking a decision on another investment.

($1 = 0.8578 euros) (Reporting by Leigh Thomas; Editing by Edmund Blair)