Geodesic
Classification of indecomposable Abelian $(v,5)$-groups
Diskretnaya Matematika, Tome 20 (2008) no. 1, pp. 94-108
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The description of the $(v,3)$-groups was given previously, it appears that all such groups are Abelian. It is known that there are non-Abelian $(v,5)$-groups. In this paper, a description of the Abelian $(v,5)$-groups is given. 
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A. A. Frolov. Classification of indecomposable Abelian $(v,5)$-groups. Diskretnaya Matematika, Tome 20 (2008) no. 1, pp. 94-108. http://geodesic.mathdoc.fr/item/DM_2008_20_1_a8/

[1] Malyshev F. M., Tarakanov V. E., “O $(v,k)$-konfiguratsiyakh”, Matem. sb., 192:9 (2001), 85–108 | MR | Zbl

[2] Trishin A. E., “Klassifikatsiya tsirkulyantnykh $(v,5)$-matrits”, Obozrenie prikladnoi i promyshlennoi matematiki, 11:2 (2004), 258–259

[3] Frolov A. A., “Opisanie abelevykh $(v,5)$-grupp”, Obozrenie prikladnoi i promyshlennoi matematiki, 11:2 (2004), 262–263