Ah, the cpuid command tells me "bytes required by XSAVE/XRSTOR area = 1088":

CPU 0:
   vendor_id = "GenuineIntel"
   version information (1/eax):
      processor type  = primary processor (0)
      family          = Intel Pentium Pro/II/III/Celeron/Core/Core 2/Atom, AMD Athlon/Duron, Cyrix M2, VIA C3 (6)
      model           = 0xe (14)
      stepping id     = 0x3 (3)
      extended family = 0x0 (0)
      extended model  = 0x5 (5)
      (simple synth)  = Intel Core i3-6000 / i5-6000 / i7-6000 / Pentium G4000 / Celeron G3900 / Xeon E3-1200 (Skylake), 14nm
   ...
   feature information (1/edx):
      FXSAVE/FXRSTOR                         = true
      ...
   feature information (1/ecx):
      XSAVE/XSTOR states                      = true
      OS-enabled XSAVE/XSTOR                  = true
      ...
   XSAVE features (0xd/0):
      bytes required by XSAVE/XRSTOR area     = 0x00000440 (1088)
      ...
   XSAVE features (0xd/1):
      XSAVEOPT instruction                        = true
      XSAVEC instruction                          = true
      XSAVES/XRSTORS instructions                 = true
      ...

/proc/cpuinfo also says:

flags		: ... xsave avx ... xsaveopt xsavec xgetbv1 xsaves ...

I recall that the Linux kernel was modified to only save AVX registers if a program uses AVX. So the process state size depends on the usage of AVX. But I cannot find the related LWN article.