According to SICSupport.co.uk which is a searchable version of the official PDF, it's:
74.10 Specialised design activities
This class includes:
- fashion design related to textiles, wearing apparel, shoes, jewellery, furniture and other interior decoration and other fashion goods as well as other personal or household goods
- industrial design, i.e. creating and developing designs and specifications that optimise the use, value and appearance of products, including the determination of the materials, mechanism, shape, colour and surface finishes of the product, taking into consideration human characteristics and needs, safety, market appeal in distribution, use and maintenance
- activities of graphic designers
This class excludes:
- design and programming of web pages, see 62.01
Not best pleased that we're an afterthought in an "other" category...
Web design is pushed over to 62 - Computer programming, consultancy and related activities, where it's not at all clear but looks like it should come under "62.01/2 Business and domestic software development", which didn't appear to be an option for me, and could come under "62.02 Computer consultancy activities"