# Group cohomology of special linear group:SL(2,5)

From Groupprops

This article gives specific information, namely, group cohomology, about a particular group, namely: special linear group:SL(2,5).

View group cohomology of particular groups | View other specific information about special linear group:SL(2,5)

This article describes the group cohomology of special linear group:SL(2,5).

## Family contexts

Family name | Parameter values | General discussion of group cohomology of family |
---|---|---|

double cover of alternating group | degree , i.e., the group | group cohomology of double cover of alternating group |

special linear group of degree two over a finite field of size | , i.e., field:F5, i.e., the group is | group cohomology of special linear group of degree two over a finite field |

## Homology groups for trivial group action

FACTS TO CHECK AGAINST(homology group for trivial group action):

First homology group: first homology group for trivial group action equals tensor product with abelianization

Second homology group: formula for second homology group for trivial group action in terms of Schur multiplier and abelianization|Hopf's formula for Schur multiplier

General: universal coefficients theorem for group homology|homology group for trivial group action commutes with direct product in second coordinate|Kunneth formula for group homology

### Over the integers

The homology groups over the integers are as follows:

The sequence of homology groups for positive degrees has a period of 4, which is in keeping with the fact that is a finite group with periodic cohomology, as are all special linear groups of degree two over a finite field.