$scope.contacts = [
      {
        "id": "1",
        "name": "first name",
        "surname": "first surname",
        "age": "29",
        "group": "group 1",
        "description": "This is first",
        "note": "some notification"
      },
      {
        "id": "2",
        "name": "second name",
        "surname": "second surname",
        "age": "27",
        "group": "group 2",
        "description": "This is first",
        "note": "some notification"
      },
      {
        "id": "3",
        "name": "third name",
        "surname": "third surname",
        "age": "25",
        "group": "group 1",
        "description": "This is first user",
        "note": "some notification"
      }]

$scope.selectGroups = new Set();
        $scope.groups = [];
         var test = [];

    for (var i = 0; i < $scope.contacts.length; i++) {
                $scope.selectGroups.add($scope.contacts[i].group);
            }

    test = JSON.stringify(Array.from($scope.selectGroups));

    $scope.groups = JSON.parse(test);

$scope.contacts.filter( contact => contact.group === 'group 1' ) //оставит кантакты только из группы 1

$scope.contacts.sort( (contact1, contact2) => contact.group.localeCompare( contact2.group ) ) //отсортирует контакты по группе.

$scope.groups = {};
    $scope.selectGroups = new Map();

             for (var i = 0; i < $scope.contacts.length; i++) {


                 if (!$scope.selectGroups.has($scope.contacts[i].group)) {
                     var users = [];
                     $scope.selectGroups.set($scope.contacts[i].group, users);

                        $scope.selectGroups.get($scope.contacts[i].group).push($scope.contacts[i].name);
                    } else {
                        $scope.selectGroups.get($scope.contacts[i].group).push($scope.contacts[i].name);
                    }

                }

$scope.groups = Array.from($scope.selectGroups);

<div ng-repeat="group in groups">

            {{group[0]}}<br>
            <div ng-repeat="gr in group[1]">
                {{gr}}
            </div>

        </div>