Evidence for an extraterrestrial flux of high-energy neutrinos has now been found in multiple searches with the IceCube detector. The first solid evidence was provided by a search for neutrino events with deposited energies $gtrsim30$ TeV and interaction vertices inside the instrumented volume. Recent analyses suggest that the extraterrestrial flux extends to lower energies and is also visible with throughgoing, $ u_mu$-induced tracks from the Northern hemisphere. Here, we combine the results from six different IceCube searches for astrophysical neutrinos in a maximum-likelihood analysis. The combined event sample features high-statistics samples of shower-like and track-like events. The data are fit in up to three observables: energy, zenith angle and event topology. Assuming the astrophysical neutrino flux to be isotropic and to consist of equal flavors at Earth, the all-flavor spectrum with neutrino energies between 25 TeV and 2.8 PeV is well described by an unbroken power law with best-fit spectral index $-2.50pm0.09$ and a flux at 100 TeV of $left(6.7_{-1.2}^{+1.1}right)cdot10^{-18},mathrm{GeV}^{-1}mathrm{s}^{-1}mathrm{sr}^{-1}mathrm{cm}^{-2}$. Under the same assumptions, an unbroken power law with index $-2$ is disfavored with a significance of 3.8 $sigma$ ($p=0.0066%$) with respect to the best fit. This significance is reduced to 2.1 $sigma$ ($p=1.7%$) if instead we compare the best fit to a spectrum with index $-2$ that has an exponential cut-off at high energies. Allowing the electron neutrino flux to deviate from the other two flavors, we find a $ u_e$ fraction of $0.18pm0.11$ at Earth. The sole production of electron neutrinos, which would be characteristic of neutron-decay dominated sources, is rejected with a significance of 3.6 $sigma$ ($p=0.014%$).
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