I am exhibiting my groups while using wp_list_categories function like so:

<?php wp_list_categories('selected=6&title_li='); ?>

Now, when hitting each category link and loading the course page, the "active" cat includes a class of "current-cat" out there element.

How do i have this current-cat class display on the specific category automatically around the index page?

I've got a category named "All" which consists of all posts (clearly) and that i want this to achieve the class "current-cat" when calling the wordpress_list_groups automatically.

Is possible?

Well the simplest solution could be CSS during my guess. When the category may be the first category within the list (first link within the list), and my prediction is it ought to be (all is most importantly, right ?) you'll be able to simply do simple CSS:

.category-widget a.current-cat, 
.home .category-widget a:first-child  {
    color:black;
}

http://www.w3.org/TR/selectors/