Solution:

Equating the two representations in base ten gives $121a + 11b + c = 512 + 64b + 8c + a,$ which simplifies to $$120a = 512 + 53b + 7c.$$ All of $a,$ $b,$ $c$ are base-eight digits, so $0 \le a, b, c \le 7$ (and $a \ge 1$ since it leads the base-eleven representation).

The right side is at least $512,$ so $a \ge 5.$ Since $N = 121a + 11b + c$ increases with $a,$ try $a = 5:$ then $53b + 7c = 88.$ Here $b = 0$ gives $7c = 88,$ impossible, and $b \ge 2$ overshoots, so $b = 1$ and $7c = 35,$ giving $c = 5.$

Thus $N = 121 \cdot 5 + 11 + 5 = 621,$ whose base-eight representation is $1155$ and base-eleven representation is $515,$ as required. The least such $N$ is $621.$