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Have they released any images of the galactic center yet?

The galactic center image won't be as fancy, unfortunately. Resolution of Webb is far-far-far too low to see anything related to the black hole, the best we could do is enhance the view of the stars near Sgr A* [1].

1. https://www.eso.org/public/videos/eso0226a/

Webb resolution is about 0.1 arc-seconds (wavelength dependent) [1]. Event Horizon Telescope (EHT) resolution is about 25 micro arc-seconds (wavelength dependent) [2].

So roughly speaking, since you cannot really compare the two, since they observe on quite different wavelengths, EHT has ~4000 times better resolution than Webb. Note that we don't speak about sensitivity here. ALMA array of radio telescopes has a collecting area of 7000 m2, while it is just one of many "nodes" of the EHT. Webb total collecting area is about 25 m2.

1. https://www.jwst.nasa.gov/content/about/faqs/faq.html#sharp

2. https://eventhorizontelescope.org/technology

Understood, but it still would be very interesting to see the area with JWST's full range of instruments.

I imagine that, along with any data for TRAPPIST, is on the [very long] list of things to do. Otherwise I'm sure it'd be posted here in no time

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