I cannot power on my iMac 27"

See if it boots to Restore Mode...

Start up from macOS Recovery

Command (⌘)-R

Install the latest macOS that was installed on your Mac.

Option-⌘-R

Upgrade to the latest macOS compatible with your Mac.

Look at top of screen for Utilities menu...

n Disk Utility>View, select Show all Devices, highlight the top left entry.

Run Disk First Aid on all items in the left panel, from top down.

Shift-Option-⌘-R

Install the macOS that came with your Mac, or the closest version still available.

Pull the power cable out of the back of the iMac for about 5 minutes. Then plug it in and try the power button. If it powers up it's probably the power supply starting to go.

If you have a 2017 iMac it is still supported by Apple and they will work on it. If it's a 2015 model Apple considered it "Vintage". 2014 or earlier is Obsolete. It will be very difficult to get parts and only Apple Authorized Service Provider will work on it.  

However, Apple will provide a free diagnostic check on any model regardless of year.

It has likely failed due to a hardware issue. IF the computer is a 2017 or older, then it is obsolete meaning parts are difficult to obtain and an Apple Store will not work on it for that reason. In that case, recycle it and replace it. However to get you started troubleshooting with Apple's advice, please click If your Mac starts up to a blank screen - Apple Support or you can simply call AppleCare for other suggestions.

AppleCare Contact Info