To see if a string is a permutation of ABCDEFG, it's easy with negative lookahead and capturing group to enforce no duplicates:
^(?!.*(.).*\1)[A-G]{7}$
You don't need the anchors if you use String.matches() in Java. Here's a test harness:
    String[] tests = {
        "ABCDEFG", // true
        "GBADFEC", // true
        "ABCADFG", // false
    };
    for (String test : tests) {
        System.out.format("%s %b%n", test,
            test.matches("(?!.*(.).*\\1)[A-G]{7}")
        );
    }
Basically, [A-G]{7}, but also (?!.*(.).*\1). That is, no character is repeated.
Here's a test harness for the assertion to play around with:
    String[] tests = {
        "abcdeb", // "(b)"
        "abcdefg", // "abcdefg"
        "aba", // "(a)"
        "abcdefgxxxhijyyy" // "(y)"
    };
    for (String test : tests) {
        System.out.println(test.replaceAll("(?=.*(.).*\\1).*", "($1)"));
    }
The way it works is by trying to match .*(.).*\1, that is, with .* in between, a captured character (.) that appears again \1.
See also