Use within the loop (on a single post) to grab all taxonomy terms for post. Take each term and add it as a custom body class.

taxonomy-term-as-class.php

//* Add taxonomy term as body class to the header
add_filter( 'body_class', 'cd_body_class' );
function cd_body_class( $classes ) {

	// Create array of terms within taxonomy named "group"
	$terms = get_the_terms( $post->ID , 'group' );
 
 	// If there are no terms, bail
 	if ( $terms == null )
 		$return;

	// Loop through terms array and assign as body class
 	foreach( $terms as $term ) {
 		$classes[] = $term->slug;
 	}

	return $classes;
}

No problem.

! $terms isn't the same as an empty() check (unless the value really is null), but if you look at the possible return values for get_the_terms(), they are array|bool|WP_Error. Looking at the function, it does its own check for an empty value and returns false, so you'll never receive an empty value. The ! $terms is just verifying that $terms isn't false.

Typically API functions should try to standardize their output and may do some error checking (and caching in this case) of their own to help keep your code a bit cleaner without having to check for more than a couple of response formats.

If figured as much regarding that return, who doesn't like extra dollars? Rather than just the return in a filter like this, you'd probably want to return the original $classes parameter so the existing body classes don't get blanked out.