With the following MySQL table:
+-----------------------------+
+ id INT UNSIGNED +
+ name VARCHAR(100) +
+----------------------------
If the query is simple and the size of returned result set is potentially large, then you may try to split it into two queries.
The first query with a narrow-down filtering criteria just to retrieve data of that row, and the second query uses COUNT with WHERE clause to calculate the position.
For example in your case
Query 1:
SELECT * FROM tbl WHERE name = 'Beta'
Query 2:
SELECT COUNT(1) FROM tbl WHERE name >= 'Beta'
We use this approach in a table with 2M record and this is way more scalable than OMG Ponies's approach.