I want to list all users with their corropsonding user class. Here are simplified versions of my tables

CREATE TABLE users (
    user_id INT NOT NULL AUTO_INCREMENT,
    user_class VARCHAR(100),
    PRIMARY KEY (user_id)
);
INSERT INTO users VALUES
    (1, '1'),
    (2, '2'),
    (3, '1,2');
CREATE TABLE classes (
    class_id INT NOT NULL AUTO_INCREMENT,
    class_name VARCHAR(100),
    PRIMARY KEY (class_id)
);
INSERT INTO classes VALUES
    (1, 'Class 1'),
    (2, 'Class 2');

And this is the query statement I am trying to use but is only returning the first matching user class and not a concatenated list as hoped.

SELECT user_id, GROUP_CONCAT(DISTINCT class_name SEPARATOR ",") AS class_name
FROM users, classes
WHERE user_class IN (class_id)
GROUP BY user_id;

Actual Output

+---------+------------+
| user_id | class_name |
+---------+------------+
|       1 | Class 1    |
|       2 | Class 2    |
|       3 | Class 1    |
+---------+------------+

Wanted Output

+---------+---------------------+
| user_id | class_name          |
+---------+---------------------+
|       1 | Class 1             |
|       2 | Class 2             |
|       3 | Class 1, Class 2    |
+---------+---------------------+

Thanks in advance