It looks like you're asking for a of a specific APK download for the Google Play Store (version 51.1 or similar) for Android.

Do not download APKs from unknown websites. Use trusted repositories:

Delete the current file, clear your browser cache, and re-download a compatible version. The Play Store Opens and Immediately Closes

In this guide, we’ll explain why a fresh APK installation improves performance, how to do it safely, and where to find the most stable version for your aging but capable device.

An updated store improves the background handshake between your OS and modern app servers, allowing you to see and download more compatible apps.

Because you are installing an app from outside the official store, you need to follow a specific sequence on Android 5.1.1. Step 1: Enable Unknown Sources

F-Droid is a fantastic app store that specializes in open-source software. If you are looking to replace essential apps without ads or tracking, F-Droid should be your first stop. It contains thousands of apps that are free as in freedom and free as in price. Because the apps on F-Droid are often lightweight and built without proprietary Google libraries, they tend to run exceptionally well on older hardware.

Running Android 5.1.1 doesn't mean you have to deal with a broken app store. By manually downloading a slightly newer, more stable Play Store APK, you can significantly enhance your device's usability, speed, and app compatibility.