How can I group posts by date on frontpage?

You test when the date changes and only show it once, right when it changed for the first time.

<?php
$date = 0;
$newDate = true;
if (have_posts()) : while (have_posts()) :

    the_post();

    if ($date == 0)
        $date = the_date();
    else if ($date != the_date()) {
        $date = the_date();
        $newDate = true;
    }

    if ($newDate)
        echo $date . ' ';

    the_title();

    $newDate = false;

    /* rest of the code */

endwhile;
endif;
?>

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