>His address as a gateway is 192.168.0.254. On it I can not intervene in any way. 1) by this do you mean you cannot change the operation *in any way* on 192.168.0.254 or you just can't change its IP address? 2) if you can change some things on .254 - can you add routes 3) if you can change some things on .254 can you add NAT entries? That is an important distinction. With any type of IP routing, both sides participating in routing have to know how to get to/from specific routes. >I think the problem is in the T2600G-28MPS My question is simple, is it possible that T2600G-28MPS (v3) does not do multivlan routing? It supports static routes - simplest form of routing there is. A better test would be to ignore the 192.168.0.254 device just for the moment. 4) can you ping devices between the different 10.10.10.0-10.10.20.0-10.10.30.0 subnets? If so then you have basic routing working in the switch. I suspect though that you may run into challenges with NAT down this path though. I can go into further detail if you want but these first four questions are important.

1-2-3 The router is fastweb and I can not do anything. I can only call them and decide the ip; 4 - the subnets and the gaetway are pinging. Arrival up to 192.168.0.252

The exact configuration is as follows:

SWITCH DOORS 1-3-5-7-9-11-13-15 - VLAN2 10.10.20.1-255.255.255.0 GW 10.10.20.1

SWITCH DOORS 2-4-6-8-10-12-14-17 VLAN3 10.10.30.1-255.255.255.0 GW 10.10.30.1

SWITCH DOORS 18-19-20-21-22-23 VLAN4 10.10.10.1-255.255.255.0 GW 10.10.10.1

DOOR SWITCH 24 VLAN1
192.168.0.252 GW 192.168.0.254

static routing 0.0.0.0 0.0.0.0 192.168.0.254

In port 15, 17, 23 I connected the respective ports of AC50 configuring him with the same three vlan

The access points take the addresses of each vlan subnet in dhcp assigned by the AC50 controller

Computers and smartphones take their addresses from the second vlan dhcp assigned by the switch and have their respective gateway (10.10.10.1)

The cable going to the router at the address 192.168.0.254 is connected to port 24.

All next hope are aimed at 192.168.0.254

The individual networks all arrive at 192.168.0.252 but do not go further. Internal internal dhcp and routing services work because I trace packets up to 192.168.0.252

Based on your post I think your switch is routing fine.
If I understand correctly, your switch is 192.168.0.252 and the 192.168.0.0/24 subnet is VLAN1.
As I said "With any type of IP routing, both sides participating in routing have to know how to get to/from specific routes. "
Your side is good with switch knowing about the subnets because they are direct connected. Unfortunately the router from fastweb does not know how to send back traffic to the switch for the 10.10.x.x subnets.
Because all of your traffic is coming from 10.10.x.x addresses the fastweb router is probably just dropping the packets in the bit bucket since it doesn't have a return route.

If you can get your 10.10.x/24 subnets added to the fastweb router via gw 192.168.2.252 then you will be able to ping the fastweb router from the AC50 and your wireless clients. BUT, that may not solve Internet Access.
I am guessing but I suspect either the FASTWEB router is performing NAT or an upstream device from FASTWEB is performing NAT for Internet access.
IMPORTANT you would ALSO have to get FASTWEB to support NAT for your 10.10.x/24 subnets.
NB: If the FASTWEB router is "kinda stupid" (or smart depending on how you look at it) it may NAT all traffic coming through it but if FASTWEB is performing NAT upstream then you may get pushback from your provider.
If FASTWEB is uncooperative *or* does not want to change anything in their router or you don't want to go back to FASTWEB then
I do see two options available to you but both have negatives.

1) You can install a firewall logically between the FASTWEB device and your subnets. Your firewalls performs the NAT and has routing entries for your various subnets
The negative is that you will have DOUBLE NAT going on. somethings: notably certain VPN protocols and some games/gaming protocols don't work or have issues with double NAT.
You will have to do some redesign but its not that hard or bad but obviously you have to acquire/deploy more hardware.
2) If you are willing to have all of your wireless clients in the 192.168.0.0/24 subnet then you do not need additional hardware and you will just make configuration changes in the AC50 and the switch.
A negative is that your wireless infrastructure will NOT have access to the Internet - some would say that is NOT a negative.
Another negative is that your wireless clients will all be in the same subnet as the FASTWEB device and you will NOT have control over DHCP assignments to them