Louis gives one number of three digits $\overline{abc}$, to their students. They had the following conversation:

$\text{Alan}:27$ divides $\overline{abc}$

$\text{Blas}:11$ divides $\overline{abc}$

$\text{David}:a+b+c=15$

$\text{Jane}:\overline{abc}$ is a divisor of $648,000$

$\text{Ronnie}:\overline{abc}$ is a perfect square

Only three of this students told the truth. Find $\overline{abc}$.

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