`$web=Get-SPWeb "http://sitecollectionurl"`
    $newGroup=$web.SiteGroups.Add($groupName,$web.Site.Owner,$web.Site.Owner,$description)
    $group=$web.SiteGroups[$groupName]
    $roleAssignment=New-Object Microsoft.SharePoint.SPRoleAssignment($group)
    $roleDefinition=$web.RoleDefinitions[$permissionLevel]
    $roleAssignment.RoleDefinitionBindings.Add($roleDefinition)
    $web.RoleAssignments.Add($roleAssignment);
    $web.Update()
    $web.Dispose()