Find domain admins group for computer

Hey,

based on a short cleanup I try to find the Domain Admins group of the domain a computer is member of with powershell.
I am trying to not use the Active Directory module since I cannot be sure it is on every computer the script will run :(
So the only technique being available is ADSI or WMI (since WMI needed hours to find the group I preferred to use ADSI).

Now my problem is:
How can I find the 'Domain Admin'/'DomainAdmins'/'Admins of Domain' or how ever the group is named?
Solution I found is SID. It seems all DomainAdmins groups have SID of S-1-5-21-*-521.

So I thought very easy ADSI filter and all is done..but this filter seems not to work correctly with SID.

'(&(objectclass=group)(objectSID=S-1-5-21-*-512))'

$domain = $(Get-WmiObject -ComputerName $env:COMPUTERNAME -Query "Select Domain from Win32_ComputerSystem").Domain

$objTrans = New-Object -comObject "NameTranslate"
$objNT =  $objTrans.GetType()
$objNT.InvokeMember("Init", "InvokeMethod",  $Null, $objTrans, (3,  $Null))
$objNT.InvokeMember("Set", "InvokeMethod",  $Null, $objTrans, (3,  $($domain.Split(".")[0] + "\")))
$LDAPDomain = "LDAP://" + $($objNT.InvokeMember("Get", "InvokeMethod",  $Null, $objTrans, 1))  

$Searcher = New-Object DirectoryServices.DirectorySearcher
$Searcher.Filter = '(&(objectclass=group)(objectSID=S-1-5-21-*-512))'
$Searcher.SearchRoot = $LDAPDomain
$Searcher.FindAll()