* Brent slightly up, WTI a fraction lower after Monday surge
* Saudi crown prince cements hold on power
* Tensions rise with Iran over Yemen conflict (Updates with comment, refreshes prices; changes dateline from TOKYO)
LONDON, Nov 7 (Reuters) - Oil eased on Tuesday, after having posted its largest gain in six weeks the previous day, while tension flared between Saudi Arabia and Iran, just as the Saudi crown prince tightened his grip on power.
Brent crude futures were down 19 cents at $64.08 a barrel by 0940 GMT, having closed 3.5 percent higher on Monday, marking the biggest percentage gain in about six weeks.
U.S. West Texas Intermediate (WTI) crude was down 7 cents at $57.28 a barrel.
Saudi Crown Prince Mohammed bin Salman moved to shore up his power base with the arrest of royals, ministers and investors, which an official described as part of "phase one" of the crackdown.
More tellingly, tensions escalated between OPEC members Saudi Arabia and Iran and it was this, more than the purge, that rattled the oil market, analysts said.
"Saudi Arabia is really going all-in again against Iran and that is for me more the focus than the domestic issue," Petromatrix strategist Olivier Jakob said.
"On the one hand, it increases the global geopolitical risk level, but it also increases the difficulty of keeping consensus within OPEC."
The Organization of the Petroleum Exporting Countries, led by Saudi Arabia, has agreed to restrain crude output by 1.8 million barrels per day together with 10 other nations including Russia until March 2018.
OPEC meets later this month and has been widely expected to extend the deal.
The Saudi-led coalition fighting against the Houthi movement in Yemen said on Monday it was closing all Yemeni air, sea and land crossings after a missile was fired towards Riyadh at the weekend.
Saudi Arabia and its Gulf allies have said they see Iran as responsible for the Yemen conflict and, on Monday Saudi Foreign Minister Adel al-Jubeir said his country reserves the right to respond to Iran's "hostile actions".
"The combined oil production of these two OPEC nations is close to 14 million bpd. When they stare at each other from very close proximity it is not surprising to see oil prices rally," PVM Oil Associates Tamas Varga said.
Even a distant threat of disruption to Middle East crude supplies from any conflict is likely to support the price, but with money managers now sitting on a record bullish bet on Brent, discord within OPEC could force their hand.
"What's happening in Saudi Arabia does not change the strategic energy priorities of the Saudis and the market is likely to dismiss this noise," UBS said in a note.
"If tensions in Yemen do not escalate, Brent prices will soon return to below $60, probably after the November OPEC meeting, which we expect to be unambitious."
(Additional reporting by Jane Chung in SEOUL and Aaron Sheldrick in TOKYO; Editing by Louise Heavens)