gistfile1.js

var context = new SP.ClientContext.get_current();

//You can do this on the current web, for this example we will be testing against a specific sub web
var targetWeb = site.openWeb(_spPageContextInfo.siteServerRelativeUrl + '/MySubWeb');
context.load(targetWeb);

var currentUser = targetWeb.get_currentUser(); 
var allGroups = targetWeb.get_siteGroups(); 
var group = allGroups.getByName('My SP Permission Group Name'); 
var groupUsers = group.get_users();

context.load(currentUser); 
context.load(allGroups); 
context.load(group); 
context.load(groupUsers);

//do the Async call to retrieve the object information (see context load)   
context.executeQueryAsync(function () {
  var valid = false; 
  var groupUserEnumerator = groupUsers.getEnumerator();

  //this group will have less than 50 people in it so enumerating the collection is deemed ok
  while (groupUserEnumerator.moveNext()) {
    var groupUser = groupUserEnumerator.get_current(); 

    if (groupUser.get_id() == currentUser.get_id()) { 
      //we have a match, the user is in the SP Group
           valid = true; 
           break; 
    } 
  }
  /* 
  The valid variable at this point tells you if the current person is in
  the SP Permissions Group
  */
  }, 
  function (sender, args) { 
     //Handle any errors with loading the SP Group     
   }
);