What is the 6-pin power connector on an HP ProDesk?

I happened to have a HP G600 standing right next to me from a customer, so I took some pictures:

As visible in the top picture, the blue and purple cables going into the 6 pin plug are not as thick as the other 4, so those would be the ones carrying the power on signal, with the 4 black and yellow ones carrying power. This is clearly not compatible with a standard PCIe 6 pin GPU connector.

I would advise you to check the HP support site for the HP ProDesk 600 G1 for a list of compatible graphics cards, that will work with your current PSU.

For example, I found this AMD Radeon HD 8490 DP (1GB) PCIe x16 Graphics Card on the list, which is a somewhat decent card, which is guaranteed to work with your system.

As mentioned in the other answers, trying to get a standard PSU to fit the case/motherboard would be a major hassle. I would definitely not recommend this.

The lesson here is: don't use a Professional Desktop PC for Home/Entertainment/Gaming purposes, if you can avoid it, as they are typically hard to upgrade.

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EDIT: After reading @Jasons comment below, I did some further research. The Q85 Chipset features one PCIe x16 expansion slot.

From the HP support web site:

Expansion slots
3 PCIe x1
1 PCIe x16

And from the Intel Q85 Chipset specifications:

Supported Processor PCI Express Port Revision         3 
Supported Processor PCI Express Port Configurations   1x16

This is obviously different from what I wrote in my hasty comment below.

Those HP monstrosities are not compatible with anything else.

You need a custom PSU. This is not going to work with a normal PSU. It's not just a matter of wiring up a conversion plug. POWER_ON line handling is different too.

Don't use the PCIe power-plug unless you like smoke to come out of the motherboard.

I would not even think about this without a ton of due diligence.

Firstly if the full sized version of the system uses the same oddball PSU as the slim versions, an ATX psu wouldn't even fit.

I'd consider a few other things - Assuming its a 12V only power input for the motherboard, a motherboard powersupply would at least have a green power on cable. Assuming one yellow 12V and one black ground cable, the other 3 cables could be power good, or sense cables or any combination. It might even be a 5V cable.

That is even assuming a PCIe cable fits (they have different 'keying' for different 6 pin molex cables), and you'd not get power anyway without the green power on cable being connected to (IIRC) ground.

In short, I suspect attempting to upgrade the PSU on anything that's not standard ATX is going to involve a ton of work with a multimeter, lots of testing and cable splicing. Even that is probably something I wouldn't particularly recommend, it could be dangerous to either you, the system or both if you get it wrong.