Creating table with plugin is not working

Your dot and quote notation is funky. Try this:

if($wpdb->get_var("SHOW TABLES LIKE '$table_name'" ) != $table_name){
    $sql= "CREATE TABLE $table_name (
           id INT(6) UNSIGNED AUTO_INCREMENT PRIMARY KEY, 
           firstname VARCHAR(30) NOT NULL,
           lastname VARCHAR(30) NOT NULL,
           email VARCHAR(50),
           reg_date TIMESTAMP
           );";
    require_once (ABSPATH. 'wp-admin/includes/upgrade.php' );
    dbDelta($sql);
}