June 21 (Reuters) - Australian shares are set to open slightly higher on Thursday, tracking improving sentiment on Wall Street as large-cap tech and consumer discretionary stocks helped lift the Nasdaq to a record close. Aussie energy stocks stand to gain from a fall in U.S. crude inventories last week which sent crude futures higher. The local share price index futures rose 0.2 percent, a 21.4-point premium to the underlying S&P/ASX 200 index close. The benchmark climbed 1.2 percent on Wednesday. New Zealand's benchmark S&P/NZX 50 index rose 0.3 percent in early trade. (Reporting by Nikhil Kurian Nainan in Bengaluru Editing by James Dalgleish)