By Nick Carey

DETROIT, May 15 (Reuters) - Nissan Motor Co has fallen behind its top North American competitors in the health of its working relationships with suppliers while General Motors Co jumped to third place in rankings released on Monday.

Toyota Motor Corp and Honda Motor Co maintained their longstanding first and second rankings in the study which looked at the six biggest automakers in North America and is closely watched in the industry.

GM's leap knocked Detroit cross-town rival Ford Motor Co into fourth place and Nissan's fall from fifth to sixth place meant Fiat Chrysler Automobiles NV (FCA) avoided coming last as it had done since 2008, despite a lower index score than in 2016.

Automakers' profitability is directly linked to good working relations with their suppliers, said study author John Henke, head of Planning Perspectives Inc, a firm that focuses on company-supplier relations.

Since the study began in 2002, Toyota has been first and Honda second on the study's OEM-Supplier Working Relations Index except for two years when the two Japanese automakers swapped the two top spots.

The study shows the results of surveys of 652 salespeople at 108 first-tier suppliers to the six automakers.

GM's rise to third place and its implications for profitability are good news for the No. 1 U.S. automaker and its chief executive Mary Barra as it wages a proxy battle with Greenlight Capital. The hedge fund has proposed splitting GM stock into two classes: one that pays dividends and one that does not.

The automaker has rejected the idea and rating agencies have said it would negatively affect GM's credit rating.

The index shows that under Barra - who took over the helm at the automaker in 2014 - GM has gone from joint last with FCA in 2015 to third place.

"If you look at the direction GM is taking, this has all happened since Mary Barra took over," Henke said. "She is changing the company."

Henke said Nissan's drop to the bottom reflected an aggressive push by the Japanese automaker to cut costs. But he said the survey results showed GM was making a similar push.

GM showed improvement across all areas included in the survey, including helpfulness and providing "profit opportunity" to suppliers. Nissan posted declines in virtually every area.

"GM is putting more pressure on their suppliers to reduce prices and improve quality than Nissan is," he said. "The question is not whether you pressure suppliers but how you do it and GM is doing it the right way." (Reporting by Nick Carey; Editing by Andrew Hay)