This is reasonably fast. It pre-generates the whole list of candidates, so it's surely not the best. The optimization of making 5 and even numbers mutually exclusive makes a huge difference.

sub perms {
    return () unless @_;
    my ($car, @cdr) = @_;
    # All permutations that do not include car
    my @cdr_perm = perms(@cdr);
    # All permutations that include car
    my @car_perm = map {
      my $this_perm = $_;
      map {
        substr($this_perm, 0, $_) . $car . substr($this_perm, $_)
      } 0..length($this_perm)
    } (@cdr_perm);
    (@car_perm, @cdr_perm, $car);
}

my $max = 0;
OUTER:
for my $num (perms(reverse(1..4,6..9)), perms(9,7,5,3,1)) {
    next unless $num > $max;
    my @digits = split //, $num;
    $num % $_ > 0 and next OUTER for @digits;
    print "$num is divisible by all of @digits\n";
    $max = $num;
}

print "Biggest found is $max\n";
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