So in the SharePoint Object Model the permissions are not members within the group. they are RoleAssignments tied to the SPList object. This is because a group can have Read access to one list and Edit to another. So you will need to look at every list and search for your group and then display its role assignment.
Try something like this and see if it works for you:
$web = get-spweb http://yourweb
$list = $web.lists["yourlist"]
$roleAssignments = $list.RoleAssignments
foreach ($oRoleAssignment in $roleAssignments)
{
foreach ($member in $oRoleAssignment.Member)
{
if ($member.name -eq "XYZ")
{
foreach ( $binding in $oRoleAssignment.RoleDefinitionBindings )
{
write-host $binding.Name
}
}
}
}
if you want to get deeper in the weeds about the SPRoledefinitionbindingcollection class check this MSDN article out:
https://msdn.microsoft.com/en-us/library/microsoft.sharepoint.sproledefinitionbindingcollection.aspx
also if you want to start adding and removing permission it will be done similar to this:
$oRoleAssignment.RoleDefinitionBindings.Remove(($web.RoleDefinitions | Where-Object { $_.Name -eq "NAME_OF_THE_PERMISSION" }))
$oRoleAssignment.Update()
I hope this helps answer your question!