TOKYO, Sept 13 (Reuters) - Shares in a Japanese foam concrete construction firm surged for the second day on Thursday on hopes that the next prime minister may hail from the family that founded -- and still runs -- the group to which it belongs.

Ruling party secretary-general Taro Aso, widely seen as a frontrunner to become prime minister after Shinzo Abe announced his resignation, is a scion of the family that founded the Aso Group, of which Aso Foam Crete Co Ltd is the only listed company.

Shares in the small, Jasdaq-traded firm, which after the stock rally has a market value of about $15 million, were up 19 percent at 464 yen by midday on Thursday.

"Basically our connection to Aso is the only reason I can see for our shares rising so much," said a company spokesman.

The Aso Group began as a coal mine in Japan's southernmost island of Kyushu in 1872 and switched to cement production after World War Two. The 60 group companies now range from the health sectors -- including a private hospital -- to trade and computer software development.

Aso ran the family concrete firm for about four years before entering parliament in 1979. His brother Yutaka runs the group.

Other Aso-linked companies only briefly benefited from their special connection after Abe's shock resignation.

Shares in Mandarake, a second-hand "manga" comic store chain, surged Wednesday on hopes that the comic-loving Aso would become premier. But by noon Thursday they were down by 6 percent at 410,000 yen.