This is similar to How to prevent system applications (like the Software Center) from asking for password? However that solution does not work for Synaptic Package Manager in 10.04.
The Software Centre falls on PolicyKit for its permissions. This allows the fine grain permissions you've seen. Synaptic does not. Like the command line tools, it needs to run as root.
The most simple way to allow a user access to it without having to type their password is by editing
You might then have to edit the launcher from